second order change of variable on y method 2

Table 2.483: second order change of variable on y method 2


#	ODE	ODE classiﬁcation	Solved?

227	$x^{2} y^{''} - 2 y^{'} x + 2 y = 0$ i.c.	[[_Emden, _Fowler], [_2nd_order, _linear, ‘_with_symmetry_[0,F(x)]‘]]	✓

228	$x^{2} y^{''} + 2 y^{'} x - 6 y = 0$ i.c.	[[_Emden, _Fowler], [_2nd_order, _linear, ‘_with_symmetry_[0,F(x)]‘]]	✓

229	$x^{2} y^{''} - y^{'} x + y = 0$ i.c.	[[_Emden, _Fowler]]	✓

230	$x^{2} y^{''} + y^{'} x + y = 0$ i.c.	[[_Emden, _Fowler], [_2nd_order, _linear, ‘_with_symmetry_[0,F(x)]‘]]	✓

244	$x^{2} y^{''} + y^{'} x - y = 0$	[[_2nd_order, _exact, _linear, _homogeneous]]	✓

245	$x^{2} y^{''} + 2 y^{'} x - 12 y = 0$	[[_Emden, _Fowler]]	✓

246	$4 x^{2} y^{''} + 8 y^{'} x - 3 y = 0$	[[_Emden, _Fowler], [_2nd_order, _linear, ‘_with_symmetry_[0,F(x)]‘]]	✓

248	$x^{2} y^{''} - 3 y^{'} x + 4 y = 0$	[[_Emden, _Fowler], [_2nd_order, _linear, ‘_with_symmetry_[0,F(x)]‘]]	✓

262	$x^{2} y^{''} - 2 y^{'} x + 2 y = 0$	[[_Emden, _Fowler], [_2nd_order, _linear, ‘_with_symmetry_[0,F(x)]‘]]	✓

315	$x^{2} y^{''} + y^{'} x + 9 y = 0$	[[_Emden, _Fowler], [_2nd_order, _linear, ‘_with_symmetry_[0,F(x)]‘]]	✓

316	$x^{2} y^{''} + 7 y^{'} x + 25 y = 0$	[[_Emden, _Fowler]]	✓

376	$x^{2} y^{''} + y^{'} x - y = 72 x^{5}$	[[_2nd_order, _exact, _linear, _nonhomogeneous]]	✓

377	$x^{2} y^{''} - 4 y^{'} x + 6 y = x^{3}$	[[_2nd_order, _with_linear_symmetries]]	✓

378	$x^{2} y^{''} - 3 y^{'} x + 4 y = x^{4}$	[[_2nd_order, _with_linear_symmetries]]	✓

379	$4 x^{2} y^{''} - 4 y^{'} x + 3 y = 8 x^{4 / 3}$	[[_2nd_order, _with_linear_symmetries]]	✓

380	$x^{2} y^{''} + y^{'} x + y = \ln (x)$	[[_2nd_order, _with_linear_symmetries]]	✓

381	$(x^{2} - 1) y^{''} - 2 y^{'} x + 2 y = x^{2} - 1$	[[_2nd_order, _with_linear_symmetries]]	✓

819	$x^{2} y^{''} - 2 y^{'} x + 2 y = 0$ i.c.	[[_Emden, _Fowler], [_2nd_order, _linear, ‘_with_symmetry_[0,F(x)]‘]]	✓

820	$x^{2} y^{''} + 2 y^{'} x - 6 y = 0$ i.c.	[[_Emden, _Fowler], [_2nd_order, _linear, ‘_with_symmetry_[0,F(x)]‘]]	✓

821	$x^{2} y^{''} - y^{'} x + y = 0$ i.c.	[[_Emden, _Fowler]]	✓

822	$x^{2} y^{''} + y^{'} x + y = 0$ i.c.	[[_Emden, _Fowler], [_2nd_order, _linear, ‘_with_symmetry_[0,F(x)]‘]]	✓

833	$x^{2} y^{''} + y^{'} x - y = 0$	[[_2nd_order, _exact, _linear, _homogeneous]]	✓

834	$x^{2} y^{''} + 2 y^{'} x - 12 y = 0$	[[_Emden, _Fowler]]	✓

835	$4 x^{2} y^{''} + 8 y^{'} x - 3 y = 0$	[[_Emden, _Fowler], [_2nd_order, _linear, ‘_with_symmetry_[0,F(x)]‘]]	✓

837	$x^{2} y^{''} - 3 y^{'} x + 4 y = 0$	[[_Emden, _Fowler], [_2nd_order, _linear, ‘_with_symmetry_[0,F(x)]‘]]	✓

860	$x^{2} y^{''} + y^{'} x + 9 y = 0$	[[_Emden, _Fowler], [_2nd_order, _linear, ‘_with_symmetry_[0,F(x)]‘]]	✓

861	$x^{2} y^{''} + 7 y^{'} x + 25 y = 0$	[[_Emden, _Fowler]]	✓

902	$x^{2} y^{''} + y^{'} x - y = 72 x^{5}$	[[_2nd_order, _exact, _linear, _nonhomogeneous]]	✓

903	$x^{2} y^{''} - 4 y^{'} x + 6 y = x^{3}$	[[_2nd_order, _with_linear_symmetries]]	✓

904	$x^{2} y^{''} - 3 y^{'} x + 4 y = x^{4}$	[[_2nd_order, _with_linear_symmetries]]	✓

905	$4 x^{2} y^{''} - 4 y^{'} x + 3 y = 8 x^{4 / 3}$	[[_2nd_order, _with_linear_symmetries]]	✓

906	$x^{2} y^{''} + y^{'} x + y = \ln (x)$	[[_2nd_order, _with_linear_symmetries]]	✓

907	$(x^{2} - 1) y^{''} - 2 y^{'} x + 2 y = x^{2} - 1$	[[_2nd_order, _with_linear_symmetries]]	✓

1293	$t^{2} y^{''} + y^{'} t + y = 0$	[[_Emden, _Fowler], [_2nd_order, _linear, ‘_with_symmetry_[0,F(x)]‘]]	✓

1294	$t^{2} y^{''} + 4 y^{'} t + 2 y = 0$	[[_2nd_order, _exact, _linear, _homogeneous]]	✓

1295	$t^{2} y^{''} + 3 y^{'} t + \frac{5 y}{4} = 0$	[[_Emden, _Fowler]]	✓

1296	$t^{2} y^{''} - 4 y^{'} t - 6 y = 0$	[[_2nd_order, _exact, _linear, _homogeneous]]	✓

1297	$t^{2} y^{''} - 4 y^{'} t + 6 y = 0$	[[_Emden, _Fowler], [_2nd_order, _linear, ‘_with_symmetry_[0,F(x)]‘]]	✓

1298	$t^{2} y^{''} - y^{'} t + 5 y = 0$	[[_Emden, _Fowler]]	✓

1299	$t^{2} y^{''} + 3 y^{'} t - 3 y = 0$	[[_Emden, _Fowler]]	✓

1300	$t^{2} y^{''} + 7 y^{'} t + 10 y = 0$	[[_Emden, _Fowler]]	✓

1327	$t^{2} y^{''} - 3 y^{'} t + 4 y = 0$	[[_Emden, _Fowler], [_2nd_order, _linear, ‘_with_symmetry_[0,F(x)]‘]]	✓

1328	$t^{2} y^{''} + 2 y^{'} t + \frac{y}{4} = 0$	[[_Emden, _Fowler]]	✓

1329	$2 t^{2} y^{''} - 5 y^{'} t + 5 y = 0$	[[_Emden, _Fowler]]	✓

1330	$t^{2} y^{''} + 3 y^{'} t + y = 0$	[[_2nd_order, _exact, _linear, _homogeneous]]	✓

1331	$4 t^{2} y^{''} - 8 y^{'} t + 9 y = 0$	[[_Emden, _Fowler]]	✓

1332	$t^{2} y^{''} + 5 y^{'} t + 13 y = 0$	[[_Emden, _Fowler]]	✓

1346	$t^{2} y^{''} - t (t + 2) y^{'} + (t + 2) y = 2 t^{3}$	[[_2nd_order, _with_linear_symmetries]]	✓

1349	$x^{2} y^{''} - 3 y^{'} x + 4 y = x^{2} \ln (x)$	[[_2nd_order, _linear, _nonhomogeneous]]	✓

1351	$t^{2} y^{''} - 2 y^{'} t + 2 y = 4 t^{2}$	[[_2nd_order, _with_linear_symmetries]]	✓

1352	$t^{2} y^{''} + 7 y^{'} t + 5 y = t$	[[_2nd_order, _exact, _linear, _nonhomogeneous]]	✓

1746	$x^{2} y^{''} + y^{'} x - y = 0$	[[_2nd_order, _exact, _linear, _homogeneous]]	✓

1747	$x^{2} y^{''} - y^{'} x + y = 0$	[[_Emden, _Fowler]]	✓

1748	$x^{2} y^{''} - (2 a - 1) x y^{'} + a^{2} y = 0$	[[_Emden, _Fowler]]	✓

1752	$4 x^{2} \sin (x) y^{''} - 4 x (\sin (x) + x \cos (x)) y^{'} + (2 x \cos (x) + 3 \sin (x)) y = 0$	[[_2nd_order, _with_linear_symmetries]]	✓

1811	$x^{2} y^{''} + y^{'} x - y = 2 x^{2} + 2$	[[_2nd_order, _exact, _linear, _nonhomogeneous]]	✓

1815	$x^{2} y^{''} - 4 y^{'} x + 6 y = x^{5 / 2}$	[[_2nd_order, _with_linear_symmetries]]	✓

1816	$x^{2} y^{''} - 3 y^{'} x + 3 y = 2 x^{4} \sin (x)$	[[_2nd_order, _with_linear_symmetries]]	✓

1820	$x^{2} y^{''} - (2 a - 1) x y^{'} + a^{2} y = x^{a + 1}$	[[_2nd_order, _with_linear_symmetries]]	✓

1828	$x^{2} y^{''} - y^{'} x - 3 y = x^{3 / 2}$	[[_2nd_order, _exact, _linear, _nonhomogeneous]]	✓

1829	$x^{2} y^{''} - x (x + 4) y^{'} + 2 (3 + x) y = x^{4} e^{x}$	[[_2nd_order, _linear, _nonhomogeneous]]	✓

1833	$4 x^{2} y^{''} - 4 x (x + 1) y^{'} + (2 x + 3) y = x^{5 / 2} e^{x}$	[[_2nd_order, _linear, _nonhomogeneous]]	✓

1838	$x^{2} y^{''} + 2 y^{'} x - 2 y = - 2 x^{2}$ i.c.	[[_2nd_order, _with_linear_symmetries]]	✓

2362	$2 t^{2} y^{''} + 3 y^{'} t - y = 0$ i.c.	[[_2nd_order, _exact, _linear, _homogeneous]]	✓

2373	$t^{2} y^{''} + α t y^{'} + β y = 0$	[[_Emden, _Fowler]]	✓

2374	$t^{2} y^{''} + 5 y^{'} t - 5 y = 0$	[[_Emden, _Fowler]]	✓

2375	$t^{2} y^{''} - y^{'} t - 2 y = 0$ i.c.	[[_Emden, _Fowler]]	✓

2385	$t^{2} y^{''} + y^{'} t + y = 0$	[[_Emden, _Fowler], [_2nd_order, _linear, ‘_with_symmetry_[0,F(x)]‘]]	✓

2386	$t^{2} y^{''} + 2 y^{'} t + 2 y = 0$	[[_Emden, _Fowler]]	✓

2395	$(- t^{2} + 1) y^{''} - 2 y^{'} t + 2 y = 0$	[_Gegenbauer]	✓

2396	$(t^{2} + 1) y^{''} - 2 y^{'} t + 2 y = 0$	[[_2nd_order, _with_linear_symmetries]]	✓

2400	$t^{2} y^{''} + 3 y^{'} t + y = 0$	[[_2nd_order, _exact, _linear, _homogeneous]]	✓

2401	$t^{2} y^{''} - y^{'} t + y = 0$	[[_Emden, _Fowler]]	✓

2411	$y^{''} - \frac{2 t y^{'}}{t^{2} + 1} + \frac{2 y}{t^{2} + 1} = t^{2} + 1$	[[_2nd_order, _with_linear_symmetries]]	✓

2431	$t^{2} y^{''} - 5 y^{'} t + 9 y = 0$	[[_Emden, _Fowler]]	✓

2432	$t^{2} y^{''} + 5 y^{'} t - 5 y = 0$	[[_Emden, _Fowler]]	✓

2433	$2 t^{2} y^{''} + 3 y^{'} t - y = 0$	[[_2nd_order, _exact, _linear, _homogeneous]]	✓

2435	$t^{2} y^{''} + 3 y^{'} t + y = 0$	[[_2nd_order, _exact, _linear, _homogeneous]]	✓

2436	$t^{2} y^{''} - y^{'} t + y = 0$	[[_Emden, _Fowler]]	✓

2438	$t^{2} y^{''} + y^{'} t + y = 0$	[[_Emden, _Fowler], [_2nd_order, _linear, ‘_with_symmetry_[0,F(x)]‘]]	✓

2439	$t^{2} y^{''} - y^{'} t + 2 y = 0$ i.c.	[[_Emden, _Fowler]]	✓

2440	$t^{2} y^{''} - 3 y^{'} t + 4 y = 0$ i.c.	[[_Emden, _Fowler], [_2nd_order, _linear, ‘_with_symmetry_[0,F(x)]‘]]	✓

2543	$2 t^{2} y^{''} + 3 y^{'} t - y = 0$ i.c.	[[_2nd_order, _exact, _linear, _homogeneous]]	✓

2554	$t^{2} y^{''} + α t y^{'} + β y = 0$	[[_Emden, _Fowler]]	✓

2555	$t^{2} y^{''} + 5 y^{'} t - 2 y = 0$ i.c.	[[_Emden, _Fowler]]	✓

2565	$t^{2} y^{''} + y^{'} t + y = 0$	[[_Emden, _Fowler], [_2nd_order, _linear, ‘_with_symmetry_[0,F(x)]‘]]	✓

2566	$t^{2} y^{''} + 2 y^{'} t + 2 y = 0$	[[_Emden, _Fowler]]	✓

2581	$t^{2} y^{''} + 3 y^{'} t + y = 0$	[[_2nd_order, _exact, _linear, _homogeneous]]	✓

2582	$t^{2} y^{''} - y^{'} t + y = 0$	[[_Emden, _Fowler]]	✓

2593	$y^{''} - \frac{2 t y^{'}}{t^{2} + 1} + \frac{2 y}{t^{2} + 1} = t^{2} + 1$	[[_2nd_order, _with_linear_symmetries]]	✓

2628	$t^{2} y^{''} + 5 y^{'} t - 5 y = 0$	[[_Emden, _Fowler]]	✓

2629	$2 t^{2} y^{''} + 3 y^{'} t - y = 0$	[[_2nd_order, _exact, _linear, _homogeneous]]	✓

2631	$t^{2} y^{''} + 3 y^{'} t + y = 0$	[[_2nd_order, _exact, _linear, _homogeneous]]	✓

2632	$t^{2} y^{''} - y^{'} t + y = 0$	[[_Emden, _Fowler]]	✓

2634	$t^{2} y^{''} + y^{'} t + y = 0$	[[_Emden, _Fowler], [_2nd_order, _linear, ‘_with_symmetry_[0,F(x)]‘]]	✓

2635	$t^{2} y^{''} + 3 y^{'} t + 2 y = 0$	[[_Emden, _Fowler]]	✓

2636	$t^{2} y^{''} - y^{'} t - 2 y = 0$ i.c.	[[_Emden, _Fowler]]	✓

2637	$t^{2} y^{''} - 3 y^{'} t + 4 y = 0$ i.c.	[[_Emden, _Fowler], [_2nd_order, _linear, ‘_with_symmetry_[0,F(x)]‘]]	✓

3221	$x^{2} y^{''} - 4 y^{'} x + y = 0$	[[_Emden, _Fowler]]	✓

3222	$x^{2} y^{''} + y^{'} x + 16 y = 0$	[[_Emden, _Fowler], [_2nd_order, _linear, ‘_with_symmetry_[0,F(x)]‘]]	✓

3223	$4 x^{2} y^{''} - 16 y^{'} x + 25 y = 0$	[[_Emden, _Fowler]]	✓

3225	$2 x^{2} y^{''} - 3 y^{'} x - 18 y = \ln (x)$	[[_2nd_order, _with_linear_symmetries]]	✓

3226	$2 x^{2} y^{''} - 3 y^{'} x + 2 y = \ln (x^{2})$	[[_2nd_order, _with_linear_symmetries]]	✓

3227	$x^{2} y^{''} - 3 y^{'} x + 4 y = x^{3}$	[[_2nd_order, _with_linear_symmetries]]	✓

3228	$x^{2} y^{''} + 3 y^{'} x + y = 1 - x$	[[_2nd_order, _exact, _linear, _nonhomogeneous]]	✓

3230	$x^{2} y^{''} - 2 y^{'} x + 2 y = 4 x + \sin (\ln (x))$	[[_2nd_order, _with_linear_symmetries]]	✓

3231	$x^{2} y^{''} - y^{'} x + 2 y = x^{2} \ln (x)$	[[_2nd_order, _linear, _nonhomogeneous]]	✓

3232	$x^{2} y^{''} + 4 y^{'} x + 3 y = (x - 1) \ln (x)$	[[_2nd_order, _linear, _nonhomogeneous]]	✓

3493	$x^{2} y^{''} - y^{'} x + y = x$	[[_2nd_order, _with_linear_symmetries]]	✓

3565	$x^{2} y^{''} + 5 y^{'} x + 3 y = 0$	[[_2nd_order, _exact, _linear, _homogeneous]]	✓

3566	$x^{2} y^{''} - 3 y^{'} x + 4 y = 0$	[[_Emden, _Fowler], [_2nd_order, _linear, ‘_with_symmetry_[0,F(x)]‘]]	✓

3567	$x^{2} y^{''} - 3 y^{'} x + 13 y = 0$	[[_Emden, _Fowler]]	✓

3568	$2 x^{2} y^{''} - y^{'} x + y = 9 x^{2}$	[[_2nd_order, _with_linear_symmetries]]	✓

3569	$x^{2} y^{''} - 4 y^{'} x + 6 y = x^{4} \sin (x)$	[[_2nd_order, _linear, _nonhomogeneous]]	✓

3575	$x^{2} y^{''} + y^{'} x - y = 0$	[[_2nd_order, _exact, _linear, _homogeneous]]	✓

3576	$x^{2} y^{''} + 5 y^{'} x + 4 y = 0$	[[_Emden, _Fowler]]	✓

3591	$x^{2} y^{''} - y^{'} x - 8 y = 0$	[[_Emden, _Fowler]]	✓

3592	$x^{2} y^{''} - 3 y^{'} x + 4 y = x^{2} \ln (x)$	[[_2nd_order, _linear, _nonhomogeneous]]	✓

3707	$x^{2} y^{''} + 3 y^{'} x - 8 y = 0$	[[_Emden, _Fowler], [_2nd_order, _linear, ‘_with_symmetry_[0,F(x)]‘]]	✓

3708	$2 x^{2} y^{''} + 5 y^{'} x + y = 0$	[[_2nd_order, _exact, _linear, _homogeneous]]	✓

3773	$x^{2} y^{''} + 4 y^{'} x + 2 y = 4 \ln (x)$	[[_2nd_order, _exact, _linear, _nonhomogeneous]]	✓

3774	$x^{2} y^{''} + 4 y^{'} x + 2 y = \cos (x)$	[[_2nd_order, _exact, _linear, _nonhomogeneous]]	✓

3775	$x^{2} y^{''} + y^{'} x + 9 y = 9 \ln (x)$	[[_2nd_order, _with_linear_symmetries]]	✓

3776	$x^{2} y^{''} - y^{'} x + 5 y = 8 x \ln {(x)}^{2}$	[[_2nd_order, _linear, _nonhomogeneous]]	✓

3777	$x^{2} y^{''} - 4 y^{'} x + 6 y = x^{4} \sin (x)$	[[_2nd_order, _linear, _nonhomogeneous]]	✓

3778	$x^{2} y^{''} + 6 y^{'} x + 6 y = 4 e^{2 x}$	[[_2nd_order, _linear, _nonhomogeneous]]	✓

3779	$x^{2} y^{''} - 3 y^{'} x + 4 y = \frac{x^{2}}{\ln (x)}$	[[_2nd_order, _linear, _nonhomogeneous]]	✓

3780	$x^{2} y^{''} - (2 m - 1) x y^{'} + m^{2} y = x^{m} \ln {(x)}^{k}$	[[_2nd_order, _linear, _nonhomogeneous]]	✓

3781	$x^{2} y^{''} - y^{'} x + 5 y = 0$ i.c.	[[_Emden, _Fowler]]	✓

3782	$t^{2} y^{''} + y^{'} t + 25 y = 0$ i.c.	[[_Emden, _Fowler], [_2nd_order, _linear, ‘_with_symmetry_[0,F(x)]‘]]	✓

4140	$x^{2} y^{''} - 2 y^{'} x + 2 y = x^{2} + 2$	[[_2nd_order, _with_linear_symmetries]]	✓

4509	$x^{2} y^{''} - y^{'} x + y = \ln (x)$	[[_2nd_order, _with_linear_symmetries]]	✓

4510	$x^{2} y^{''} + 3 y^{'} x + 5 y = \frac{5 \ln (x)}{x^{2}}$	[[_2nd_order, _linear, _nonhomogeneous]]	✓

5990	$x^{2} y^{''} - y^{'} x + y = x$	[[_2nd_order, _with_linear_symmetries]]	✓

5991	$y^{''} - \frac{2 y^{'}}{x} + \frac{2 y}{x^{2}} = x \ln (x)$	[[_2nd_order, _exact, _linear, _nonhomogeneous]]	✓

5992	$x^{2} y^{''} + y^{'} x - 4 y = x^{3}$	[[_2nd_order, _with_linear_symmetries]]	✓

5993	$x^{2} y^{''} + y^{'} x - y = x^{2} e^{- x}$	[[_2nd_order, _exact, _linear, _nonhomogeneous]]	✓

5994	$2 x^{2} y^{''} + 3 y^{'} x - y = \frac{1}{x}$	[[_2nd_order, _exact, _linear, _nonhomogeneous]]	✓

6026	$x^{2} y^{''} - 2 y^{'} x + 2 y = 0$	[[_Emden, _Fowler], [_2nd_order, _linear, ‘_with_symmetry_[0,F(x)]‘]]	✓

6192	$x^{2} y^{''} + 3 y^{'} x - 3 y = 0$	[[_Emden, _Fowler]]	✓

6193	$x^{2} y^{''} + y^{'} x - 4 y = 0$	[[_Emden, _Fowler], [_2nd_order, _linear, ‘_with_symmetry_[0,F(x)]‘]]	✓

6194	$x^{2} y^{''} + 7 y^{'} x + 9 y = 0$	[[_Emden, _Fowler]]	✓

6196	$x^{2} y^{''} + y^{'} x - 16 y = 8 x^{4}$	[[_2nd_order, _with_linear_symmetries]]	✓

6197	$x^{2} y^{''} + y^{'} x - y = x - \frac{1}{x}$	[[_2nd_order, _exact, _linear, _nonhomogeneous]]	✓

6198	$x^{2} y^{''} - 5 y^{'} x + 9 y = 2 x^{3}$	[[_2nd_order, _with_linear_symmetries]]	✓

6199	$x^{2} y^{''} - 3 y^{'} x + 4 y = 6 x^{2} \ln (x)$	[[_2nd_order, _linear, _nonhomogeneous]]	✓

6201	$x^{2} y^{''} + y^{'} x + y = 2 x$	[[_2nd_order, _with_linear_symmetries]]	✓

6215	$x^{2} y^{''} - y^{'} x + y = x$	[[_2nd_order, _with_linear_symmetries]]	✓

6249	$x^{2} y^{''} - 3 y^{'} x + 3 y = 0$	[[_Emden, _Fowler]]	✓

6251	$(x^{2} + 2 x) y^{''} - 2 (x + 1) y^{'} + 2 y = 0$	[[_2nd_order, _with_linear_symmetries]]	✓

6253	$(x^{2} + 1) y^{''} - 2 y^{'} x + 2 y = 0$	[[_2nd_order, _with_linear_symmetries]]	✓

6408	$x (1 - x) y^{''} + 2 (- 2 x + 1) y^{'} - 2 y = 0$	[[_2nd_order, _exact, _linear, _homogeneous]]	✓

6409	$x^{2} y^{''} + y^{'} x - 9 y = 0$	[[_Emden, _Fowler], [_2nd_order, _linear, ‘_with_symmetry_[0,F(x)]‘]]	✓

6411	$x^{2} y^{''} - y^{'} x + y = 0$	[[_Emden, _Fowler]]	✓

6532	$t^{2} N^{''} - 2 t N^{'} + 2 N = t \ln (t)$	[[_2nd_order, _with_linear_symmetries]]	✓

6540	$y^{''} + \frac{y^{'}}{x} - \frac{y}{x^{2}} = \ln (x)$	[[_2nd_order, _exact, _linear, _nonhomogeneous]]	✓

6695	$x^{2} y^{''} - 3 y^{'} x + 4 y = 0$	[[_Emden, _Fowler], [_2nd_order, _linear, ‘_with_symmetry_[0,F(x)]‘]]	✓

6749	$x^{2} y^{''} - 3 y^{'} x + 4 y = x + x^{2} \ln (x)$	[[_2nd_order, _linear, _nonhomogeneous]]	✓

6750	$x^{2} y^{''} - 2 y^{'} x + 2 y = \ln {(x)}^{2} - \ln (x^{2})$	[[_2nd_order, _with_linear_symmetries]]	✓

6756	$(x^{2} + 1) y^{''} - 2 y^{'} x + 2 y = 2$	[[_2nd_order, _with_linear_symmetries]]	✓

6757	$(x^{2} + 4) y^{''} - 2 y^{'} x + 2 y = 8$	[[_2nd_order, _with_linear_symmetries]]	✓

6766	$(x \sin (x) + \cos (x)) y^{''} - x \cos (x) y^{'} + \cos (x) y = x$	[[_2nd_order, _with_linear_symmetries]]	✓

6767	$x y^{''} - 3 y^{'} + \frac{3 y}{x} = x + 2$	[[_2nd_order, _with_linear_symmetries]]	✓

6771	$(x^{2} + 1) y^{''} - 2 y^{'} x + 2 y = \frac{- x^{2} + 1}{x}$	[[_2nd_order, _with_linear_symmetries]]	✓

6912	$x^{2} y^{''} - 7 y^{'} x + 15 y = 0$	[[_Emden, _Fowler]]	✓

6998	$x^{2} y^{''} + y^{'} x + y = 0$	[[_Emden, _Fowler], [_2nd_order, _linear, ‘_with_symmetry_[0,F(x)]‘]]	✓

6999	$x^{2} y^{''} + y^{'} x + y = \sec (\ln (x))$	[[_2nd_order, _linear, _nonhomogeneous]]	✓

7479	$y^{''} + \frac{y^{'}}{x} - \frac{y}{x^{2}} = 0$	[[_2nd_order, _exact, _linear, _homogeneous]]	✓

7484	$(- x^{2} + 1) y^{''} - y^{'} x + y = 0$	[[_2nd_order, _exact, _linear, _homogeneous]]	✓

7487	$x^{2} y^{''} + 2 y^{'} x + 4 y = 0$	[[_Emden, _Fowler]]	✓

7492	$x^{2} y^{''} + y^{'} x - y = x^{2} + 2 x$	[[_2nd_order, _exact, _linear, _nonhomogeneous]]	✓

7494	$x (x + 1) y^{''} + (x + 2) y^{'} - y = x + \frac{1}{x}$	[[_2nd_order, _exact, _linear, _nonhomogeneous]]	✓

7497	$x^{2} (\ln (x) - 1) y^{''} - y^{'} x + y = x {(1 - \ln (x))}^{2}$	[[_2nd_order, _with_linear_symmetries]]	✓

7524	$p x^{2} u^{''} + q x u^{'} + r u = f (x)$	[[_2nd_order, _linear, _nonhomogeneous]]	✓

7527	$y^{''} - \frac{x y^{'}}{- x^{2} + 1} + \frac{y}{- x^{2} + 1} = 0$	[_Gegenbauer, [_2nd_order, _linear, ‘_with_symmetry_[0,F(x)]‘]]	✓

7674	$y^{''} + \frac{y^{'}}{x} - \frac{y}{x^{2}} = 0$ i.c.	[[_2nd_order, _exact, _linear, _homogeneous]]	✓

7675	$y^{''} + \frac{y^{'}}{x} - \frac{y}{x^{2}} = 0$ i.c.	[[_2nd_order, _exact, _linear, _homogeneous]]	✓

7699	$x^{2} y^{''} + 2 y^{'} x - 6 y = 0$	[[_Emden, _Fowler], [_2nd_order, _linear, ‘_with_symmetry_[0,F(x)]‘]]	✓

7700	$2 x^{2} y^{''} + y^{'} x - y = 0$	[[_Emden, _Fowler]]	✓

7701	$x^{2} y^{''} + y^{'} x - 4 y = 0$	[[_Emden, _Fowler], [_2nd_order, _linear, ‘_with_symmetry_[0,F(x)]‘]]	✓

7702	$x^{2} y^{''} - 5 y^{'} x + 9 y = x^{2}$	[[_2nd_order, _with_linear_symmetries]]	✓

7704	$x^{2} y^{''} + y^{'} x + 4 y = 1$	[[_2nd_order, _with_linear_symmetries]]	✓

7705	$x^{2} y^{''} - 3 y^{'} x + 5 y = 0$	[[_Emden, _Fowler]]	✓

7706	$x^{2} y^{''} + (- 2 - i) x y^{'} + 3 i y = 0$	[[_Emden, _Fowler]]	✓

7707	$x^{2} y^{''} + y^{'} x - 4 π y = x$	[[_2nd_order, _with_linear_symmetries]]	✓

7961	$x^{2} y^{''} + 3 y^{'} x + 10 y = 0$	[[_Emden, _Fowler]]	✓

7962	$2 x^{2} y^{''} + 10 y^{'} x + 8 y = 0$	[[_Emden, _Fowler]]	✓

7963	$x^{2} y^{''} + 2 y^{'} x - 12 y = 0$	[[_Emden, _Fowler]]	✓

7965	$x^{2} y^{''} - 3 y^{'} x + 4 y = 0$	[[_Emden, _Fowler], [_2nd_order, _linear, ‘_with_symmetry_[0,F(x)]‘]]	✓

7966	$x^{2} y^{''} + 2 y^{'} x - 6 y = 0$	[[_Emden, _Fowler], [_2nd_order, _linear, ‘_with_symmetry_[0,F(x)]‘]]	✓

7967	$x^{2} y^{''} + 2 y^{'} x + 3 y = 0$	[[_Emden, _Fowler]]	✓

7968	$x^{2} y^{''} + y^{'} x - 2 y = 0$	[[_Emden, _Fowler], [_2nd_order, _linear, ‘_with_symmetry_[0,F(x)]‘]]	✓

7969	$x^{2} y^{''} + y^{'} x - 16 y = 0$	[[_Emden, _Fowler], [_2nd_order, _linear, ‘_with_symmetry_[0,F(x)]‘]]	✓

8000	$(x^{2} - 1) y^{''} - 2 y^{'} x + 2 y = {(x^{2} - 1)}^{2}$	[[_2nd_order, _with_linear_symmetries]]	✓

8004	$x^{2} y^{''} - 2 y^{'} x + 2 y = x e^{- x}$	[[_2nd_order, _with_linear_symmetries]]	✓

8061	$x^{2} y^{''} + 3 y^{'} x + y = \frac{2}{x}$	[[_2nd_order, _exact, _linear, _nonhomogeneous]]	✓

8606	$2 x^{2} y^{''} + y^{'} x - y = 0$	[[_Emden, _Fowler]]	✓

8607	$2 x^{2} y^{''} - 3 y^{'} x + 2 y = 0$	[[_Emden, _Fowler], [_2nd_order, _linear, ‘_with_symmetry_[0,F(x)]‘]]	✓

8609	$2 x^{2} y^{''} + 5 y^{'} x - 2 y = 0$	[[_Emden, _Fowler], [_2nd_order, _linear, ‘_with_symmetry_[0,F(x)]‘]]	✓


8610	$x^{2} y^{''} + 2 y^{'} x - 12 y = 0$	[[_Emden, _Fowler]]	✓

8611	$x^{2} y^{''} + y^{'} x - 9 y = 0$	[[_Emden, _Fowler], [_2nd_order, _linear, ‘_with_symmetry_[0,F(x)]‘]]	✓

8612	$x^{2} y^{''} - 3 y^{'} x + 4 y = 0$	[[_Emden, _Fowler], [_2nd_order, _linear, ‘_with_symmetry_[0,F(x)]‘]]	✓

8613	$x^{2} y^{''} - 5 y^{'} x + 9 y = 0$	[[_Emden, _Fowler]]	✓

8614	$x^{2} y^{''} + 5 y^{'} x + 5 y = 0$	[[_Emden, _Fowler]]	✓

8761	$t^{2} y^{''} - 3 y^{'} t + 5 y = 0$	[[_Emden, _Fowler]]	✓

8873	$x^{4} y^{''} + x^{3} y^{'} - 4 x^{2} y = 1$	[[_2nd_order, _with_linear_symmetries]]	✓

8874	$x^{4} y^{''} + x^{3} y^{'} - 4 x^{2} y = x$	[[_2nd_order, _with_linear_symmetries]]	✓

8875	$x^{2} y^{''} + y^{'} x - 4 y = x$	[[_2nd_order, _with_linear_symmetries]]	✓

9142	$x^{2} y^{''} - 3 y^{'} x + 3 y = 2 x^{3} - x^{2}$	[[_2nd_order, _with_linear_symmetries]]	✓

9169	$x^{2} y^{''} - 4 y^{'} x + 6 y = 0$	[[_Emden, _Fowler], [_2nd_order, _linear, ‘_with_symmetry_[0,F(x)]‘]]	✓

11141	$x^{2} y^{''} + y^{'} x - y - a x^{2} = 0$	[[_2nd_order, _exact, _linear, _nonhomogeneous]]	✓

11142	$x^{2} y^{''} + y^{'} x + a y = 0$	[[_Emden, _Fowler], [_2nd_order, _linear, ‘_with_symmetry_[0,F(x)]‘]]	✓

11148	$x^{2} y^{''} - y^{'} x + y - 3 x^{3} = 0$	[[_2nd_order, _with_linear_symmetries]]	✓

11156	$x^{2} y^{''} - 2 y^{'} x + 2 y - x^{5} \ln (x) = 0$	[[_2nd_order, _with_linear_symmetries]]	✓

11157	$x^{2} y^{''} - 2 y^{'} x - 4 y - x \sin (x) - (a x^{2} + 12 a + 4) \cos (x) = 0$	[[_2nd_order, _exact, _linear, _nonhomogeneous]]	✓

11164	$x^{2} y^{''} - 3 y^{'} x + 4 y - 5 x = 0$	[[_2nd_order, _with_linear_symmetries]]	✓

11165	$x^{2} y^{''} - 3 y^{'} x - 5 y - x^{2} \ln (x) = 0$	[[_2nd_order, _exact, _linear, _nonhomogeneous]]	✓

11166	$x^{2} y^{''} - 4 y^{'} x + 6 y - x^{4} + x^{2} = 0$	[[_2nd_order, _linear, _nonhomogeneous]]	✓

11168	$x^{2} y^{''} - 5 y^{'} x + 8 y - \sin (x) x^{3} = 0$	[[_2nd_order, _linear, _nonhomogeneous]]	✓

11169	$x^{2} y^{''} + a x y^{'} + b y = 0$	[[_Emden, _Fowler]]	✓

11206	$(x^{2} + 1) y^{''} - y^{'} x + y = 0$	[[_2nd_order, _with_linear_symmetries]]	✓

11208	$(x^{2} + 1) y^{''} - 2 y^{'} x + 2 y = 0$	[[_2nd_order, _with_linear_symmetries]]	✓

11233	$x (x + 1) y^{''} + (3 x + 2) y^{'} + y = 0$	[[_2nd_order, _exact, _linear, _homogeneous]]	✓

11259	$4 x^{2} y^{''} + 5 y^{'} x - y - \ln (x) = 0$	[[_2nd_order, _with_linear_symmetries]]	✓

11279	$(a^{2} x^{2} - 1) y^{''} + 2 a^{2} x y^{'} - 2 a^{2} y = 0$	[_Gegenbauer]	✓

11280	$(a x^{2} + b x) y^{''} + 2 b y^{'} - 2 a y = 0$	[[_2nd_order, _exact, _linear, _homogeneous]]	✓

11287	$x^{3} y^{''} - x^{2} y^{'} + x y - \ln {(x)}^{3} = 0$	[[_2nd_order, _with_linear_symmetries]]	✓

11289	$x^{3} y^{''} + 3 x^{2} y^{'} + x y - 1 = 0$	[[_2nd_order, _with_linear_symmetries]]	✓

11300	$x^{2} (x + 1) y^{''} - x (2 x + 1) y^{'} + (2 x + 1) y = 0$	[[_2nd_order, _with_linear_symmetries]]	✓

11302	$y^{''} = - \frac{2 (x - 2) y^{'}}{x (x - 1)} + \frac{2 (x + 1) y}{x^{2} (x - 1)}$	[[_2nd_order, _with_linear_symmetries], [_2nd_order, _linear, ‘_with_symmetry_[0,F(x)]‘]]	✓

11303	$y^{''} = \frac{(5 x - 4) y^{'}}{x (x - 1)} - \frac{(9 x - 6) y}{x^{2} (x - 1)}$	[[_2nd_order, _with_linear_symmetries]]	✓

11310	$y^{''} = \frac{(x - 4) y^{'}}{2 x (x - 2)} - \frac{(x - 3) y}{2 x^{2} (x - 2)}$	[[_2nd_order, _with_linear_symmetries]]	✓

11311	$y^{''} = \frac{y^{'}}{x + 1} - \frac{(3 x + 1) y}{4 x^{2} (x + 1)}$	[[_2nd_order, _with_linear_symmetries]]	✓

11319	$y^{''} = \frac{2 (a x + 2 b) y^{'}}{x (a x + b)} - \frac{(2 a x + 6 b) y}{(a x + b) x^{2}}$	[[_2nd_order, _with_linear_symmetries]]	✓

11337	$y^{''} = \frac{(x^{2} - 2) y^{'}}{x (x^{2} - 1)} - \frac{(x^{2} - 2) y}{x^{2} (x^{2} - 1)}$	[[_2nd_order, _with_linear_symmetries]]	✓

11370	$y^{''} = \frac{(7 a x^{2} + 5) y^{'}}{x (a x^{2} + 1)} - \frac{(15 a x^{2} + 5) y}{x^{2} (a x^{2} + 1)}$	[[_2nd_order, _with_linear_symmetries]]	✓

11392	$y^{''} = \frac{y^{'}}{x (\ln (x) - 1)} - \frac{y}{x^{2} (\ln (x) - 1)}$	[[_2nd_order, _with_linear_symmetries]]	✓

11397	$y^{''} = - \frac{x \sin (x) y^{'}}{x \cos (x) - \sin (x)} + \frac{\sin (x) y}{x \cos (x) - \sin (x)}$	[[_2nd_order, _with_linear_symmetries]]	✓

11398	$y^{''} = - \frac{(x^{2} \sin (x) - 2 x \cos (x)) y^{'}}{x^{2} \cos (x)} - \frac{(2 \cos (x) - x \sin (x)) y}{x^{2} \cos (x)}$	[[_2nd_order, _with_linear_symmetries]]	✓

12544	$x^{2} y^{''} + a x y^{'} + b y = 0$	[[_Emden, _Fowler]]	✓

12613	$x^{2} (a x + b) y^{''} - 2 x (a x + 2 b) y^{'} + 2 (a x + 3 b) y = 0$	[[_2nd_order, _with_linear_symmetries]]	✓

12879	$x^{2} y^{''} + 3 y^{'} x + y = \frac{1}{{(1 - x)}^{2}}$	[[_2nd_order, _exact, _linear, _nonhomogeneous]]	✓

12896	$(x^{2} + 1) y^{''} + 2 y^{'} x - 2 y = 0$	[[_2nd_order, _with_linear_symmetries]]	✓

12910	$(x^{2} + 1) y^{''} - 2 y^{'} x + 2 y = 0$	[[_2nd_order, _with_linear_symmetries]]	✓

12913	$(2 x^{3} - 1) y^{''} - 6 x^{2} y^{'} + 6 x y = 0$	[[_2nd_order, _with_linear_symmetries]]	✓

12929	$x^{2} y^{''} + 3 y^{'} x + y = x$	[[_2nd_order, _exact, _linear, _nonhomogeneous]]	✓

13085	$t^{2} x^{''} + 3 x^{'} t + x = 0$	[[_2nd_order, _exact, _linear, _homogeneous]]	✓

13086	$t x^{''} + 4 x^{'} + \frac{2 x}{t} = 0$	[[_Emden, _Fowler], [_2nd_order, _linear, ‘_with_symmetry_[0,F(x)]‘]]	✓

13087	$t^{2} x^{''} - 7 x^{'} t + 16 x = 0$	[[_Emden, _Fowler]]	✓

13088	$t^{2} x^{''} + 3 x^{'} t - 8 x = 0$ i.c.	[[_Emden, _Fowler], [_2nd_order, _linear, ‘_with_symmetry_[0,F(x)]‘]]	✓

13090	$t^{2} x^{''} - x^{'} t + 2 x = 0$ i.c.	[[_Emden, _Fowler]]	✓

13099	$t^{2} x^{''} - 3 x^{'} t + 3 x = 4 t^{7}$	[[_2nd_order, _with_linear_symmetries]]	✓

13323	$x^{2} y^{''} - 2 y^{'} x + 2 y = 0$ i.c.	[[_Emden, _Fowler], [_2nd_order, _linear, ‘_with_symmetry_[0,F(x)]‘]]	✓

13324	$x^{2} y^{''} + y^{'} x - 4 y = 0$ i.c.	[[_Emden, _Fowler], [_2nd_order, _linear, ‘_with_symmetry_[0,F(x)]‘]]	✓

13452	$x^{2} y^{''} - 6 y^{'} x + 10 y = 3 x^{4} + 6 x^{3}$	[[_2nd_order, _linear, _nonhomogeneous]]	✓

13454	$(x^{2} + 2 x) y^{''} - 2 (x + 1) y^{'} + 2 y = {(x + 2)}^{2}$	[[_2nd_order, _with_linear_symmetries]]	✓

13455	$x^{2} y^{''} - x (x + 2) y^{'} + (x + 2) y = x^{3}$	[[_2nd_order, _with_linear_symmetries]]	✓

13457	$(2 x + 1) (x + 1) y^{''} + 2 y^{'} x - 2 y = {(2 x + 1)}^{2}$	[[_2nd_order, _exact, _linear, _nonhomogeneous]]	✓

13460	$x^{2} y^{''} - 3 y^{'} x + 3 y = 0$	[[_Emden, _Fowler]]	✓

13461	$x^{2} y^{''} + y^{'} x - 4 y = 0$	[[_Emden, _Fowler], [_2nd_order, _linear, ‘_with_symmetry_[0,F(x)]‘]]	✓

13462	$4 x^{2} y^{''} - 4 y^{'} x + 3 y = 0$	[[_Emden, _Fowler], [_2nd_order, _linear, ‘_with_symmetry_[0,F(x)]‘]]	✓

13463	$x^{2} y^{''} - 3 y^{'} x + 4 y = 0$	[[_Emden, _Fowler], [_2nd_order, _linear, ‘_with_symmetry_[0,F(x)]‘]]	✓

13464	$x^{2} y^{''} + y^{'} x + 4 y = 0$	[[_Emden, _Fowler], [_2nd_order, _linear, ‘_with_symmetry_[0,F(x)]‘]]	✓

13465	$x^{2} y^{''} - 3 y^{'} x + 13 y = 0$	[[_Emden, _Fowler]]	✓

13466	$3 x^{2} y^{''} - 4 y^{'} x + 2 y = 0$	[[_Emden, _Fowler], [_2nd_order, _linear, ‘_with_symmetry_[0,F(x)]‘]]	✓

13467	$x^{2} y^{''} + y^{'} x + 9 y = 0$	[[_Emden, _Fowler], [_2nd_order, _linear, ‘_with_symmetry_[0,F(x)]‘]]	✓

13468	$9 x^{2} y^{''} + 3 y^{'} x + y = 0$	[[_Emden, _Fowler], [_2nd_order, _linear, ‘_with_symmetry_[0,F(x)]‘]]	✓

13469	$x^{2} y^{''} - 5 y^{'} x + 10 y = 0$	[[_Emden, _Fowler]]	✓

13473	$x^{2} y^{''} - 4 y^{'} x + 6 y = 4 x - 6$	[[_2nd_order, _with_linear_symmetries]]	✓

13474	$x^{2} y^{''} - 5 y^{'} x + 8 y = 2 x^{3}$	[[_2nd_order, _with_linear_symmetries]]	✓

13475	$x^{2} y^{''} + 4 y^{'} x + 2 y = 4 \ln (x)$	[[_2nd_order, _exact, _linear, _nonhomogeneous]]	✓

13476	$x^{2} y^{''} + y^{'} x + 4 y = 2 x \ln (x)$	[[_2nd_order, _with_linear_symmetries]]	✓

13477	$x^{2} y^{''} + y^{'} x + 4 y = 4 \sin (\ln (x))$	[[_2nd_order, _linear, _nonhomogeneous]]	✓

13479	$x^{2} y^{''} - 2 y^{'} x - 10 y = 0$ i.c.	[[_Emden, _Fowler], [_2nd_order, _linear, ‘_with_symmetry_[0,F(x)]‘]]	✓

13480	$x^{2} y^{''} - 4 y^{'} x + 6 y = 0$ i.c.	[[_Emden, _Fowler], [_2nd_order, _linear, ‘_with_symmetry_[0,F(x)]‘]]	✓

13481	$x^{2} y^{''} + 5 y^{'} x + 3 y = 0$ i.c.	[[_2nd_order, _exact, _linear, _homogeneous]]	✓

13483	$x^{2} y^{''} - 4 y^{'} x + 4 y = - 6 x^{3} + 4 x^{2}$ i.c.	[[_2nd_order, _with_linear_symmetries]]	✓

13484	$x^{2} y^{''} + 2 y^{'} x - 6 y = 10 x^{2}$ i.c.	[[_2nd_order, _with_linear_symmetries]]	✓

13485	$x^{2} y^{''} - 5 y^{'} x + 8 y = 2 x^{3}$ i.c.	[[_2nd_order, _with_linear_symmetries]]	✓

13601	$\frac{(1 + t) x^{''}}{t} - \frac{x^{'}}{t^{2}} + \frac{x}{t^{3}} = 0$	[[_2nd_order, _with_linear_symmetries]]	✓

13602	$t^{2} x^{''} + x^{'} t + x = 0$	[[_Emden, _Fowler], [_2nd_order, _linear, ‘_with_symmetry_[0,F(x)]‘]]	✓

13611	$x y^{''} + y^{'} + \frac{λ y}{x} = 0$ i.c.	[[_Emden, _Fowler], [_2nd_order, _linear, ‘_with_symmetry_[0,F(x)]‘]]	✓

13612	$x y^{''} + y^{'} + \frac{λ y}{x} = 0$ i.c.	[[_Emden, _Fowler], [_2nd_order, _linear, ‘_with_symmetry_[0,F(x)]‘]]	✓

13725	$x^{2} y^{''} - 4 y^{'} x + 6 y = 0$ i.c.	[[_Emden, _Fowler], [_2nd_order, _linear, ‘_with_symmetry_[0,F(x)]‘]]	✓

13727	$t^{2} x^{''} - 5 x^{'} t + 10 x = 0$ i.c.	[[_Emden, _Fowler]]	✓

13728	$t^{2} x^{''} + x^{'} t - x = 0$ i.c.	[[_2nd_order, _exact, _linear, _homogeneous]]	✓

13730	$x^{2} y^{''} - y^{'} x - 3 y = 0$ i.c.	[[_2nd_order, _exact, _linear, _homogeneous]]	✓

13731	$4 t^{2} x^{''} + 8 x^{'} t + 5 x = 0$ i.c.	[[_Emden, _Fowler]]	✓

13732	$x^{2} y^{''} - 5 y^{'} x + 5 y = 0$ i.c.	[[_Emden, _Fowler]]	✓

13733	$3 x^{2} z^{''} + 5 x z^{'} - z = 0$ i.c.	[[_2nd_order, _exact, _linear, _homogeneous]]	✓

13833	$x^{2} y^{''} - 4 y^{'} x + 6 y = 2$	[[_2nd_order, _with_linear_symmetries]]	✓

13942	$y^{''} + \frac{y^{'}}{x + 1} - \frac{(x + 2) y}{x^{2} (x + 1)} = 0$	[[_2nd_order, _with_linear_symmetries]]	✓

14016	$t^{2} y^{''} + 3 y^{'} t + y = t^{7}$	[[_2nd_order, _exact, _linear, _nonhomogeneous]]	✓

14239	$x^{2} y^{''} + y^{'} x - y = 0$	[[_2nd_order, _exact, _linear, _homogeneous]]	✓

14241	$2 x^{2} y^{''} + 3 y^{'} x - y = 0$	[[_2nd_order, _exact, _linear, _homogeneous]]	✓

14249	$2 x^{2} y^{''} + y^{'} x - y = 0$	[[_Emden, _Fowler]]	✓

14257	$x^{2} y^{''} + 6 y^{'} x + 4 y = 0$	[[_2nd_order, _exact, _linear, _homogeneous]]	✓

14258	$x^{2} y^{''} - 5 y^{'} x + 9 y = 0$	[[_Emden, _Fowler]]	✓

14274	$x^{2} y^{''} - 4 y^{'} x + 6 y = 0$ i.c.	[[_Emden, _Fowler], [_2nd_order, _linear, ‘_with_symmetry_[0,F(x)]‘]]	✓

14275	$x^{2} y^{''} - 4 y^{'} x + 6 y = 0$ i.c.	[[_Emden, _Fowler], [_2nd_order, _linear, ‘_with_symmetry_[0,F(x)]‘]]	✓

14276	$x^{2} y^{''} - 4 y^{'} x + 6 y = 0$ i.c.	[[_Emden, _Fowler], [_2nd_order, _linear, ‘_with_symmetry_[0,F(x)]‘]]	✓

14277	$x^{2} y^{''} - 4 y^{'} x + 6 y = 0$ i.c.	[[_Emden, _Fowler], [_2nd_order, _linear, ‘_with_symmetry_[0,F(x)]‘]]	✓

14278	$x^{2} y^{''} - 4 y^{'} x + 6 y = 0$ i.c.	[[_Emden, _Fowler], [_2nd_order, _linear, ‘_with_symmetry_[0,F(x)]‘]]	✓

14417	$x^{2} y^{''} + 2 y^{'} x - 2 y = 0$	[[_Emden, _Fowler]]	✓

14421	$x^{2} y^{''} - y^{'} x + y = 0$ i.c.	[[_Emden, _Fowler]]	✓

14424	$x^{2} y^{''} + y^{'} x - 4 y = - 3 x - \frac{3}{x}$ i.c.	[[_2nd_order, _with_linear_symmetries]]	✓

15229	$x^{2} y^{''} - 4 y^{'} x + 6 y = 0$ i.c.	[[_Emden, _Fowler], [_2nd_order, _linear, ‘_with_symmetry_[0,F(x)]‘]]	✓

15230	$4 x^{2} y^{''} + 4 y^{'} x - y = 0$ i.c.	[[_Emden, _Fowler], [_2nd_order, _linear, ‘_with_symmetry_[0,F(x)]‘]]	✓

15231	$x^{2} y^{''} - y^{'} x + y = 0$ i.c.	[[_Emden, _Fowler]]	✓

15234	$x^{2} y^{''} - 4 y^{'} x + 6 y = 0$	[[_Emden, _Fowler], [_2nd_order, _linear, ‘_with_symmetry_[0,F(x)]‘]]	✓

15308	$x^{2} y^{''} - 5 y^{'} x + 8 y = 0$	[[_Emden, _Fowler], [_2nd_order, _linear, ‘_with_symmetry_[0,F(x)]‘]]	✓

15311	$2 x^{2} y^{''} - y^{'} x + y = 0$	[[_Emden, _Fowler], [_2nd_order, _linear, ‘_with_symmetry_[0,F(x)]‘]]	✓

15312	$x^{2} y^{''} - 5 y^{'} x + 9 y = 0$	[[_Emden, _Fowler]]	✓

15313	$x^{2} y^{''} + 5 y^{'} x + 4 y = 0$	[[_Emden, _Fowler]]	✓

15315	$x^{2} y^{''} - 19 y^{'} x + 100 y = 0$	[[_Emden, _Fowler]]	✓

15317	$x^{2} y^{''} - y^{'} x + 10 y = 0$	[[_Emden, _Fowler]]	✓

15318	$x^{2} y^{''} + 5 y^{'} x + 29 y = 0$	[[_Emden, _Fowler]]	✓

15319	$x^{2} y^{''} + y^{'} x + y = 0$	[[_Emden, _Fowler], [_2nd_order, _linear, ‘_with_symmetry_[0,F(x)]‘]]	✓

15320	$2 x^{2} y^{''} + 5 y^{'} x + y = 0$	[[_2nd_order, _exact, _linear, _homogeneous]]	✓

15323	$x^{2} y^{''} + y^{'} x - 25 y = 0$	[[_Emden, _Fowler], [_2nd_order, _linear, ‘_with_symmetry_[0,F(x)]‘]]	✓

15324	$4 x^{2} y^{''} + 8 y^{'} x + 5 y = 0$	[[_Emden, _Fowler]]	✓

15325	$3 x^{2} y^{''} - 7 y^{'} x + 3 y = 0$	[[_Emden, _Fowler]]	✓

15326	$x^{2} y^{''} - 2 y^{'} x - 10 y = 0$ i.c.	[[_Emden, _Fowler], [_2nd_order, _linear, ‘_with_symmetry_[0,F(x)]‘]]	✓

15327	$4 x^{2} y^{''} + 4 y^{'} x - y = 0$ i.c.	[[_Emden, _Fowler], [_2nd_order, _linear, ‘_with_symmetry_[0,F(x)]‘]]	✓

15328	$x^{2} y^{''} - 11 y^{'} x + 36 y = 0$ i.c.	[[_Emden, _Fowler]]	✓

15329	$x^{2} y^{''} - y^{'} x + y = 0$ i.c.	[[_Emden, _Fowler]]	✓

15330	$x^{2} y^{''} - y^{'} x + 2 y = 0$ i.c.	[[_Emden, _Fowler]]	✓

15331	$x^{2} y^{''} - 3 y^{'} x + 13 y = 0$ i.c.	[[_Emden, _Fowler]]	✓

15348	$x^{2} y^{''} - 4 y^{'} x + 6 y = 10 x + 12$ i.c.	[[_2nd_order, _with_linear_symmetries]]	✓

15354	$x^{2} y^{''} - 4 y^{'} x + 6 y = 1$	[[_2nd_order, _with_linear_symmetries]]	✓

15355	$x^{2} y^{''} - 4 y^{'} x + 6 y = x$	[[_2nd_order, _with_linear_symmetries]]	✓

15356	$x^{2} y^{''} - 4 y^{'} x + 6 y = 22 x + 24$	[[_2nd_order, _with_linear_symmetries]]	✓

15357	$x^{2} y^{''} - 7 y^{'} x + 15 y = x^{2}$	[[_2nd_order, _with_linear_symmetries]]	✓

15358	$x^{2} y^{''} - 7 y^{'} x + 15 y = x$	[[_2nd_order, _with_linear_symmetries]]	✓

15359	$x^{2} y^{''} - 7 y^{'} x + 15 y = 1$	[[_2nd_order, _with_linear_symmetries]]	✓

15360	$x^{2} y^{''} - 7 y^{'} x + 15 y = 4 x^{2} + 2 x + 3$	[[_2nd_order, _with_linear_symmetries]]	✓

15434	$x^{2} y^{''} - 5 y^{'} x + 8 y = \frac{5}{x^{3}}$	[[_2nd_order, _with_linear_symmetries]]	✓

15435	$2 x^{2} y^{''} - y^{'} x + y = \frac{50}{x^{3}}$	[[_2nd_order, _with_linear_symmetries]]	✓

15436	$2 x^{2} y^{''} + 5 y^{'} x + y = 85 \cos (2 \ln (x))$	[[_2nd_order, _exact, _linear, _nonhomogeneous]]	✓

15438	$3 x^{2} y^{''} - 7 y^{'} x + 3 y = 4 x^{3}$	[[_2nd_order, _with_linear_symmetries]]	✓

15439	$2 x^{2} y^{''} + 5 y^{'} x + y = \frac{10}{x}$	[[_2nd_order, _exact, _linear, _nonhomogeneous]]	✓

15440	$x^{2} y^{''} - 5 y^{'} x + 9 y = 6 x^{3}$	[[_2nd_order, _with_linear_symmetries]]	✓

15441	$x^{2} y^{''} + 5 y^{'} x + 4 y = 64 x^{2} \ln (x)$	[[_2nd_order, _linear, _nonhomogeneous]]	✓

15442	$x^{2} y^{''} - 2 y^{'} x + 2 y = 3 \sqrt{x}$	[[_2nd_order, _with_linear_symmetries]]	✓

15448	$x^{2} y^{''} + y^{'} x - y = \sqrt{x}$	[[_2nd_order, _exact, _linear, _nonhomogeneous]]	✓

15449	$x^{2} y^{''} + y^{'} x - 9 y = 12 x^{3}$	[[_2nd_order, _with_linear_symmetries]]	✓

15450	$x^{2} y^{''} - 3 y^{'} x + 4 y = x^{2}$	[[_2nd_order, _with_linear_symmetries]]	✓

15451	$x^{2} y^{''} + 5 y^{'} x + 4 y = \ln (x)$	[[_2nd_order, _with_linear_symmetries]]	✓

15456	$x^{2} y^{''} - 2 y^{'} x - 4 y = \frac{10}{x}$ i.c.	[[_2nd_order, _exact, _linear, _nonhomogeneous]]	✓

15466	$x^{2} y^{''} + y^{'} x - 9 y = 0$	[[_Emden, _Fowler], [_2nd_order, _linear, ‘_with_symmetry_[0,F(x)]‘]]	✓

15469	$x^{2} y^{''} - 7 y^{'} x + 16 y = 0$	[[_Emden, _Fowler]]	✓

15474	$x^{2} y^{''} + 7 y^{'} x + 9 y = 0$	[[_Emden, _Fowler]]	✓

15480	$x^{2} y^{''} + y^{'} x + 9 y = 0$	[[_Emden, _Fowler], [_2nd_order, _linear, ‘_with_symmetry_[0,F(x)]‘]]	✓

15482	$x^{2} y^{''} + 2 y^{'} x - 30 y = 0$	[[_Emden, _Fowler]]	✓

15485	$4 x^{2} y^{''} + 8 y^{'} x + y = 0$	[[_Emden, _Fowler]]	✓

15487	$2 x^{2} y^{''} - 3 y^{'} x + 2 y = 0$	[[_Emden, _Fowler], [_2nd_order, _linear, ‘_with_symmetry_[0,F(x)]‘]]	✓

15488	$9 x^{2} y^{''} + 3 y^{'} x + y = 0$	[[_Emden, _Fowler], [_2nd_order, _linear, ‘_with_symmetry_[0,F(x)]‘]]	✓

15498	$x^{2} y^{''} + y^{'} x - 9 y = 3 \sqrt{x}$	[[_2nd_order, _with_linear_symmetries]]	✓

15501	$x^{2} y^{''} + 2 y^{'} x - 6 y = 18 \ln (x)$	[[_2nd_order, _with_linear_symmetries]]	✓

15503	$2 x^{2} y^{''} - y^{'} x - 2 y = 10 x^{2}$	[[_2nd_order, _with_linear_symmetries]]	✓

15506	$x^{2} y^{''} + 3 y^{'} x + 2 y = 6$	[[_2nd_order, _with_linear_symmetries]]	✓

15507	$x^{2} y^{''} + y^{'} x - y = \frac{1}{x^{2} + 1}$	[[_2nd_order, _exact, _linear, _nonhomogeneous]]	✓

15512	$x^{2} y^{''} + 3 y^{'} x + y = \frac{1}{{(x + 1)}^{2}}$	[[_2nd_order, _exact, _linear, _nonhomogeneous]]	✓

15513	$x^{2} y^{''} + 3 y^{'} x + y = \frac{1}{x}$	[[_2nd_order, _exact, _linear, _nonhomogeneous]]	✓

15729	$x^{2} y^{''} - 12 y^{'} x + 42 y = 0$	[[_Emden, _Fowler]]	✓

15730	$t^{2} y^{''} + 3 y^{'} t + 5 y = 0$	[[_Emden, _Fowler]]	✓

15755	$t^{2} y^{''} - 12 y^{'} t + 42 y = 0$ i.c.	[[_Emden, _Fowler]]	✓

15756	$x^{2} y^{''} + 3 y^{'} x + 5 y = 0$ i.c.	[[_Emden, _Fowler]]	✓

15775	$x^{2} y^{''} - y^{'} x - 16 y = 0$	[[_Emden, _Fowler]]	✓

15776	$x^{2} y^{''} + 3 y^{'} x + 2 y = 0$	[[_Emden, _Fowler]]	✓

15787	$x^{2} y^{''} + 5 y^{'} x + 4 y = 0$ i.c.	[[_Emden, _Fowler]]	✓

15930	$y^{''} - \frac{y^{'}}{t} + \frac{y}{t^{2}} = \frac{1}{t}$	[[_2nd_order, _exact, _linear, _nonhomogeneous]]	✓

16108	$2 t^{2} y^{''} - 3 y^{'} t - 3 y = 0$	[[_Emden, _Fowler]]	✓

16112	$3 t^{2} y^{''} - 5 y^{'} t - 3 y = 0$ i.c.	[[_Emden, _Fowler]]	✓

16113	$t^{2} y^{''} + 7 y^{'} t - 7 y = 0$ i.c.	[[_Emden, _Fowler]]	✓

16118	$t^{2} y^{''} + y^{'} t - y = 0$	[[_2nd_order, _exact, _linear, _homogeneous]]	✓

16130	$t^{2} y^{''} + a t y^{'} + b y = 0$	[[_Emden, _Fowler]]	✓

16169	$3 t^{2} y^{''} - 2 y^{'} t + 2 y = 0$	[[_Emden, _Fowler]]	✓

16170	$t^{2} y^{''} - y^{'} t + y = 0$	[[_Emden, _Fowler]]	✓

16283	$t^{2} y^{''} + 3 y^{'} t + y = \ln (t)$	[[_2nd_order, _exact, _linear, _nonhomogeneous]]	✓

16284	$t^{2} y^{''} + y^{'} t + 4 y = t$	[[_2nd_order, _with_linear_symmetries]]	✓

16285	$t^{2} y^{''} - 4 y^{'} t - 6 y = 2 \ln (t)$	[[_2nd_order, _exact, _linear, _nonhomogeneous]]	✓

16294	$t^{2} (\ln (t) - 1) y^{''} - y^{'} t + y = - \frac{3 (1 + \ln (t))}{4 \sqrt{t}}$ i.c.	[[_2nd_order, _with_linear_symmetries]]	✓

16295	$(\sin (t) - t \cos (t)) y^{''} - t \sin (t) y^{'} + \sin (t) y = t$ i.c.	[[_2nd_order, _with_linear_symmetries]]	✓

16369	$4 x^{2} y^{''} - 8 y^{'} x + 5 y = 0$	[[_Emden, _Fowler], [_2nd_order, _linear, ‘_with_symmetry_[0,F(x)]‘]]	✓

16370	$3 x^{2} y^{''} - 4 y^{'} x + 2 y = 0$	[[_Emden, _Fowler], [_2nd_order, _linear, ‘_with_symmetry_[0,F(x)]‘]]	✓

16371	$2 x^{2} y^{''} - 8 y^{'} x + 8 y = 0$	[[_Emden, _Fowler]]	✓

16372	$2 x^{2} y^{''} - 7 y^{'} x + 7 y = 0$	[[_Emden, _Fowler]]	✓

16374	$9 x^{2} y^{''} - 9 y^{'} x + 10 y = 0$	[[_Emden, _Fowler]]	✓

16375	$2 x^{2} y^{''} - 2 y^{'} x + 20 y = 0$	[[_Emden, _Fowler]]	✓

16376	$x^{2} y^{''} - 5 y^{'} x + 10 y = 0$	[[_Emden, _Fowler]]	✓

16377	$4 x^{2} y^{''} + 8 y^{'} x + y = 0$	[[_Emden, _Fowler]]	✓

16379	$x^{2} y^{''} - 5 y^{'} x + 9 y = 0$	[[_Emden, _Fowler]]	✓

16380	$x^{2} y^{''} + 7 y^{'} x + 9 y = 0$	[[_Emden, _Fowler]]	✓

16389	$x^{2} y^{''} + 5 y^{'} x + 4 y = \frac{1}{x^{5}}$	[[_2nd_order, _with_linear_symmetries]]	✓

16390	$x^{2} y^{''} - 5 y^{'} x + 9 y = x^{3}$	[[_2nd_order, _with_linear_symmetries]]	✓

16391	$x^{2} y^{''} + y^{'} x + y = \frac{1}{x^{2}}$	[[_2nd_order, _with_linear_symmetries]]	✓

16392	$x^{2} y^{''} + y^{'} x + 4 y = \frac{1}{x^{2}}$	[[_2nd_order, _with_linear_symmetries]]	✓

16393	$x^{2} y^{''} + 2 y^{'} x - 6 y = 2 x$	[[_2nd_order, _with_linear_symmetries]]	✓

16394	$x^{2} y^{''} + y^{'} x - 16 y = \ln (x)$	[[_2nd_order, _with_linear_symmetries]]	✓

16395	$x^{2} y^{''} + y^{'} x + 4 y = 8$	[[_2nd_order, _with_linear_symmetries]]	✓

16396	$x^{2} y^{''} + y^{'} x + 36 y = x^{2}$	[[_2nd_order, _with_linear_symmetries]]	✓

16399	$3 x^{2} y^{''} - 4 y^{'} x + 2 y = 0$ i.c.	[[_Emden, _Fowler], [_2nd_order, _linear, ‘_with_symmetry_[0,F(x)]‘]]	✓

16400	$2 x^{2} y^{''} - 7 y^{'} x + 7 y = 0$ i.c.	[[_Emden, _Fowler]]	✓

16401	$x^{2} y^{''} + y^{'} x + 4 y = 0$ i.c.	[[_Emden, _Fowler], [_2nd_order, _linear, ‘_with_symmetry_[0,F(x)]‘]]	✓

16407	$2 x^{2} y^{''} + 3 y^{'} x - y = \frac{1}{x^{2}}$ i.c.	[[_2nd_order, _exact, _linear, _nonhomogeneous]]	✓

16408	$x^{2} y^{''} + 4 y^{'} x + 2 y = \ln (x)$ i.c.	[[_2nd_order, _exact, _linear, _nonhomogeneous]]	✓

16410	$9 x^{2} y^{''} + 27 y^{'} x + 10 y = \frac{1}{x}$ i.c.	[[_2nd_order, _with_linear_symmetries]]	✓

16411	$x^{2} y^{''} - y^{'} x + 2 y = 0$	[[_Emden, _Fowler]]	✓

16412	$x^{2} y^{''} + 4 y^{'} x + 2 y = 0$	[[_2nd_order, _exact, _linear, _homogeneous]]	✓

16413	$x^{2} y^{''} + y^{'} x + y = 0$	[[_Emden, _Fowler], [_2nd_order, _linear, ‘_with_symmetry_[0,F(x)]‘]]	✓

16425	$x^{2} y^{''} + 4 y^{'} x + 2 y = 0$	[[_2nd_order, _exact, _linear, _homogeneous]]	✓

16426	$x^{2} y^{''} + y^{'} x + y = x^{2}$	[[_2nd_order, _with_linear_symmetries]]	✓

16427	$x^{2} y^{''} + y^{'} x + 4 y = 0$ i.c.	[[_Emden, _Fowler], [_2nd_order, _linear, ‘_with_symmetry_[0,F(x)]‘]]	✓

16428	$x^{2} y^{''} - y^{'} x + y = 0$ i.c.	[[_Emden, _Fowler]]	✓

16435	$6 x^{2} y^{''} + 5 y^{'} x - y = 0$ i.c.	[[_Emden, _Fowler], [_2nd_order, _linear, ‘_with_symmetry_[0,F(x)]‘]]	✓

16535	$t^{2} y^{''} - 5 y^{'} t + 5 y = 0$	[[_Emden, _Fowler]]	✓

16536	$x^{2} y^{''} + 7 y^{'} x + 8 y = 0$	[[_Emden, _Fowler]]	✓

16537	$x^{2} y^{''} - 4 y^{'} x + 6 y = 0$	[[_Emden, _Fowler], [_2nd_order, _linear, ‘_with_symmetry_[0,F(x)]‘]]	✓

16538	$x^{2} y^{''} + y^{'} x + y = 0$	[[_Emden, _Fowler], [_2nd_order, _linear, ‘_with_symmetry_[0,F(x)]‘]]	✓

16539	$2 x^{2} y^{''} + 5 y^{'} x + y = 0$ i.c.	[[_2nd_order, _exact, _linear, _homogeneous]]	✓

16540	$5 x^{2} y^{''} - y^{'} x + 2 y = 0$	[[_Emden, _Fowler]]	✓

16541	$x^{2} y^{''} - 7 y^{'} x + 25 y = 0$	[[_Emden, _Fowler]]	✓

16542	$x^{2} y^{''} - 7 y^{'} x + 15 y = 8 x$	[[_2nd_order, _with_linear_symmetries]]	✓

17046	$x^{2} y^{''} + y^{'} x - y = 0$	[[_2nd_order, _exact, _linear, _homogeneous]]	✓

17047	$x^{2} y^{''} + 3 y^{'} x + y = 0$	[[_2nd_order, _exact, _linear, _homogeneous]]	✓

17048	$x^{2} y^{''} + 2 y^{'} x + 6 y = 0$	[[_Emden, _Fowler]]	✓

17056	$x^{2} y^{''} + y^{'} x + y = x (6 - \ln (x))$	[[_2nd_order, _with_linear_symmetries]]	✓

17058	$x^{2} y^{''} - y^{'} x - 3 y = - \frac{16 \ln (x)}{x}$	[[_2nd_order, _exact, _linear, _nonhomogeneous]]	✓

17059	$x^{2} y^{''} - 2 y^{'} x - 2 y = x^{2} - 2 x + 2$	[[_2nd_order, _with_linear_symmetries]]	✓

17060	$x^{2} y^{''} + y^{'} x - y = x^{m}$	[[_2nd_order, _exact, _linear, _nonhomogeneous]]	✓

17061	$x^{2} y^{''} + 4 y^{'} x + 2 y = 2 \ln {(x)}^{2} + 12 x$	[[_2nd_order, _exact, _linear, _nonhomogeneous]]	✓

17065	$(x^{2} - x) y^{''} + (2 x - 3) y^{'} - 2 y = 0$	[_Jacobi]	✓

17066	$(2 x^{2} + 3 x) y^{''} - 6 (x + 1) y^{'} + 6 y = 6$	[[_2nd_order, _with_linear_symmetries]]	✓

17095	$2 x^{2} (2 - \ln (x)) y^{''} + x (4 - \ln (x)) y^{'} - y = \frac{{(2 - \ln (x))}^{2}}{\sqrt{x}}$ i.c.	[[_2nd_order, _linear, _nonhomogeneous]]	✓

17097	$x^{3} (\ln (x) - 1) y^{''} - x^{2} y^{'} + x y = 2 \ln (x)$ i.c.	[[_2nd_order, _with_linear_symmetries]]	✓

17480	$a x^{2} y^{''} + b x y^{'} + c y = d$	[[_2nd_order, _with_linear_symmetries]]	✓

17498	$x^{2} y^{''} - x (x + 2) y^{'} + (x + 2) y = 0$	[[_2nd_order, _with_linear_symmetries]]	✓

17499	$(1 - x \cot (x)) y^{''} - y^{'} x + y = 0$	[[_2nd_order, _with_linear_symmetries]]	✓

17555	$a x^{2} y^{''} + b x y^{'} + c y = 0$	[[_Emden, _Fowler]]	✓

17556	$x^{2} y^{''} + y^{'} x + 4 y = 0$	[[_Emden, _Fowler], [_2nd_order, _linear, ‘_with_symmetry_[0,F(x)]‘]]	✓

17557	$x^{2} y^{''} + 4 y^{'} x + 2 y = 0$	[[_2nd_order, _exact, _linear, _homogeneous]]	✓

17558	$x^{2} y^{''} + 3 y^{'} x + \frac{5 y}{4} = 0$	[[_Emden, _Fowler]]	✓

17559	$x^{2} y^{''} - 4 y^{'} x - 6 y = 0$	[[_2nd_order, _exact, _linear, _homogeneous]]	✓

17561	$x^{2} y^{''} - 5 y^{'} x + 9 y = 0$	[[_Emden, _Fowler]]	✓

17562	$x^{2} y^{''} + 2 y^{'} x + 4 y = 0$	[[_Emden, _Fowler]]	✓

17563	$2 x^{2} y^{''} - 4 y^{'} x + 6 y = 0$	[[_Emden, _Fowler]]	✓

17564	$2 x^{2} y^{''} + y^{'} x - 3 y = 0$ i.c.	[[_2nd_order, _exact, _linear, _homogeneous]]	✓

17565	$4 x^{2} y^{''} + 8 y^{'} x + 17 y = 0$ i.c.	[[_Emden, _Fowler]]	✓

17566	$x^{2} y^{''} - 5 y^{'} x + 9 y = 0$ i.c.	[[_Emden, _Fowler]]	✓

17567	$x^{2} y^{''} + 3 y^{'} x + 5 y = 0$ i.c.	[[_Emden, _Fowler]]	✓

17601	$x^{2} y^{''} - 3 y^{'} x + 4 y = \ln (x)$	[[_2nd_order, _with_linear_symmetries]]	✓

17602	$x^{2} y^{''} + 7 y^{'} x + 5 y = x$	[[_2nd_order, _exact, _linear, _nonhomogeneous]]	✓

17603	$x^{2} y^{''} - 2 y^{'} x + 2 y = 3 x^{2} + 2 \ln (x)$	[[_2nd_order, _with_linear_symmetries]]	✓

17604	$x^{2} y^{''} + y^{'} x + 4 y = \sin (\ln (x))$	[[_2nd_order, _linear, _nonhomogeneous]]	✓

17628	$t^{2} y^{''} - t (t + 2) y^{'} + (t + 2) y = 2 t^{3}$	[[_2nd_order, _with_linear_symmetries]]	✓

17635	$x^{2} y^{''} - 3 y^{'} x + 4 y = x^{2} \ln (x)$	[[_2nd_order, _linear, _nonhomogeneous]]	✓

17636	$t^{2} y^{''} - 2 y^{'} t + 2 y = 4 t^{2}$	[[_2nd_order, _with_linear_symmetries]]	✓

17637	$t^{2} y^{''} + 7 y^{'} t + 5 y = t$	[[_2nd_order, _exact, _linear, _nonhomogeneous]]	✓

17930	$x^{2} y^{''} - 2 y^{'} x + 2 y = 2 x^{3}$	[[_2nd_order, _with_linear_symmetries]]	✓

17955	$y^{''} + \frac{2 y^{'}}{x} - \frac{n (n + 1) y}{x^{2}} = 0$	[[_2nd_order, _with_linear_symmetries]]	✓

17956	$x^{2} y^{''} - 4 y^{'} x + 6 y = x$	[[_2nd_order, _with_linear_symmetries]]	✓

17957	$x^{2} y^{''} - y^{'} x + 2 y = x \ln (x)$	[[_2nd_order, _with_linear_symmetries]]	✓

18181	$x^{3} y^{''} + x^{2} y^{'} + x y = 1$	[[_2nd_order, _with_linear_symmetries]]	✓

18189	$x^{2} y^{''} - 3 y^{'} x - 5 y = 0$	[[_2nd_order, _exact, _linear, _homogeneous]]	✓

18192	$x^{2} y^{''} - 2 y^{'} x + 2 y = 0$ i.c.	[[_Emden, _Fowler], [_2nd_order, _linear, ‘_with_symmetry_[0,F(x)]‘]]	✓

18241	$x^{2} y^{''} + 3 y^{'} x + 10 y = 0$	[[_Emden, _Fowler]]	✓

18242	$2 x^{2} y^{''} + 10 y^{'} x + 8 y = 0$	[[_Emden, _Fowler]]	✓

18243	$x^{2} y^{''} + 2 y^{'} x - 12 y = 0$	[[_Emden, _Fowler]]	✓

18245	$x^{2} y^{''} - 3 y^{'} x + 4 y = 0$	[[_Emden, _Fowler], [_2nd_order, _linear, ‘_with_symmetry_[0,F(x)]‘]]	✓

18246	$x^{2} y^{''} + 2 y^{'} x - 6 y = 0$	[[_Emden, _Fowler], [_2nd_order, _linear, ‘_with_symmetry_[0,F(x)]‘]]	✓

18247	$x^{2} y^{''} + 2 y^{'} x + 3 y = 0$	[[_Emden, _Fowler]]	✓

18248	$x^{2} y^{''} + y^{'} x - 2 y = 0$	[[_Emden, _Fowler], [_2nd_order, _linear, ‘_with_symmetry_[0,F(x)]‘]]	✓

18249	$x^{2} y^{''} + y^{'} x - 16 y = 0$	[[_Emden, _Fowler], [_2nd_order, _linear, ‘_with_symmetry_[0,F(x)]‘]]	✓

18281	$(x^{2} - 1) y^{''} - 2 y^{'} x + 2 y = {(x^{2} - 1)}^{2}$	[[_2nd_order, _with_linear_symmetries]]	✓

18285	$x^{2} y^{''} - 2 y^{'} x + 2 y = x e^{- x}$	[[_2nd_order, _with_linear_symmetries]]	✓

18442	$t^{2} x^{''} - 6 x^{'} t + 12 x = 0$	[[_Emden, _Fowler]]	✓

18445	$t^{2} x^{''} - 2 x^{'} t + 2 x = 0$	[[_Emden, _Fowler], [_2nd_order, _linear, ‘_with_symmetry_[0,F(x)]‘]]	✓

18524	$x^{2} y^{''} + 3 y^{'} x + y = \frac{1}{x}$	[[_2nd_order, _exact, _linear, _nonhomogeneous]]	✓

18535	$x^{2} y^{''} - 5 y^{'} x + 5 y = \frac{1}{x}$	[[_2nd_order, _with_linear_symmetries]]	✓

18544	$x^{2} y^{''} + y^{'} x + y = 0$	[[_Emden, _Fowler], [_2nd_order, _linear, ‘_with_symmetry_[0,F(x)]‘]]	✓

18614	$x^{2} y^{''} + 3 y^{'} x - 8 y = x$	[[_2nd_order, _with_linear_symmetries]]	✓

18619	$x^{2} y^{''} - y^{'} x + y = \ln (x)$	[[_2nd_order, _with_linear_symmetries]]	✓

18623	$(x^{2} - x) y^{''} + (3 x - 2) y^{'} + y = 0$	[[_2nd_order, _exact, _linear, _homogeneous]]	✓

18624	$(3 x^{2} + x) y^{''} + 2 (1 + 6 x) y^{'} + 6 y = \sin (x)$	[[_2nd_order, _exact, _linear, _nonhomogeneous]]	✓

18653	$y^{''} - \frac{2 y^{'}}{x} + \frac{2 y}{x^{2}} = 0$	[[_2nd_order, _exact, _linear, _homogeneous]]	✓

18852	$x^{2} y^{''} - y^{'} x + y = 2 \ln (x)$	[[_2nd_order, _with_linear_symmetries]]	✓

18856	$x^{2} y^{''} - 2 y^{'} x - 4 y = x^{4}$	[[_2nd_order, _exact, _linear, _nonhomogeneous]]	✓

18857	$x^{2} y^{''} + 5 y^{'} x + 4 y = x^{4}$	[[_2nd_order, _with_linear_symmetries]]	✓

18858	$x^{2} y^{''} + 2 y^{'} x - 20 y = {(x + 1)}^{2}$	[[_2nd_order, _with_linear_symmetries]]	✓

18859	$x^{2} y^{''} + 7 y^{'} x + 5 y = x^{5}$	[[_2nd_order, _exact, _linear, _nonhomogeneous]]	✓

18863	$x^{2} y^{''} + 4 y^{'} x + 2 y = e^{x}$	[[_2nd_order, _exact, _linear, _nonhomogeneous]]	✓

18869	$x^{2} y^{''} + y^{'} x - y = x^{m}$	[[_2nd_order, _exact, _linear, _nonhomogeneous]]	✓

18870	$x^{2} y^{''} - 3 y^{'} x + 4 y = x^{m}$	[[_2nd_order, _with_linear_symmetries]]	✓

18873	$x^{2} y^{''} + 3 y^{'} x + y = \frac{1}{{(1 - x)}^{2}}$	[[_2nd_order, _exact, _linear, _nonhomogeneous]]	✓

18874	$x^{2} y^{''} - (2 m - 1) x y^{'} + (m^{2} + n^{2}) y = n^{2} x^{m} \ln (x)$	[[_2nd_order, _linear, _nonhomogeneous]]	✓

18875	$x^{2} y^{''} - 3 y^{'} x + y = \frac{\ln (x) \sin (\ln (x)) + 1}{x}$	[[_2nd_order, _linear, _nonhomogeneous]]	✓

18935	$x^{2} y^{''} + y^{'} x - y = 0$	[[_2nd_order, _exact, _linear, _homogeneous]]	✓

18954	$(- x^{2} + 1) y^{''} + y^{'} x - y = x {(- x^{2} + 1)}^{3 / 2}$	[[_2nd_order, _with_linear_symmetries]]	✓

18974	$x^{2} y^{''} - 5 y^{'} x + 5 y = \frac{1}{x}$	[[_2nd_order, _with_linear_symmetries]]	✓

19244	$x^{2} y^{''} + 2 y^{'} x - 2 y = 0$	[[_Emden, _Fowler]]	✓

19245	$x^{2} y^{''} - y^{'} x + y = 2 \ln (x)$	[[_2nd_order, _with_linear_symmetries]]	✓

19252	$x^{2} y^{''} - y^{'} x + 5 y = 0$	[[_Emden, _Fowler]]	✓

19255	$x^{2} y^{''} + 7 y^{'} x + 5 y = x^{5}$	[[_2nd_order, _exact, _linear, _nonhomogeneous]]	✓

19256	$x^{2} y^{''} + 5 y^{'} x + 4 y = x^{4}$	[[_2nd_order, _with_linear_symmetries]]	✓

19257	$x^{2} y^{''} - 4 y^{'} x + 6 y = x^{4}$	[[_2nd_order, _with_linear_symmetries]]	✓

19258	$x^{2} y^{''} - 2 y^{'} x - 4 y = x^{4}$	[[_2nd_order, _exact, _linear, _nonhomogeneous]]	✓

19259	$x^{2} y^{''} + y^{'} x - y = x^{m}$	[[_2nd_order, _exact, _linear, _nonhomogeneous]]	✓

19260	$x^{2} y^{''} - 3 y^{'} x + 4 y = x^{m}$	[[_2nd_order, _with_linear_symmetries]]	✓

19262	$x^{2} y^{''} + 4 y^{'} x + 2 y = e^{x}$	[[_2nd_order, _exact, _linear, _nonhomogeneous]]	✓

19263	$x^{2} y^{''} + 3 y^{'} x - 3 y = x$	[[_2nd_order, _with_linear_symmetries]]	✓

19267	$x^{2} y^{''} + 2 y^{'} x - 20 y = {(x + 1)}^{2}$	[[_2nd_order, _with_linear_symmetries]]	✓

19270	$x^{2} y^{''} - y^{'} x + 2 y = x \ln (x)$	[[_2nd_order, _with_linear_symmetries]]	✓

19271	$x^{2} y^{''} - 3 y^{'} x + 5 y = x^{2} \sin (\ln (x))$	[[_2nd_order, _linear, _nonhomogeneous]]	✓

19284	$(- x^{2} + 1) y^{''} - y^{'} x + y = 2 x$	[[_2nd_order, _exact, _linear, _nonhomogeneous]]	✓

19287	$(- b x^{2} + a x) y^{''} + 2 a y^{'} + 2 b y = 0$	[[_2nd_order, _exact, _linear, _homogeneous]]	✓

19412	$(x^{2} + a) y^{''} - 2 y^{'} x + 2 y = 0$	[[_2nd_order, _with_linear_symmetries]]	✓

19419	$x^{2} y^{''} + y^{'} x - y = 8 x^{3}$	[[_2nd_order, _exact, _linear, _nonhomogeneous]]	✓

19425	$x^{2} y^{''} + y^{'} x - y = 0$	[[_2nd_order, _exact, _linear, _homogeneous]]	✓

19435	$(- x^{2} + 1) y^{''} + y^{'} x - y = x {(- x^{2} + 1)}^{3 / 2}$	[[_2nd_order, _with_linear_symmetries]]	✓

19436	$x^{2} y^{''} - (x^{2} + 2 x) y^{'} + (x + 2) y = 0$	[[_2nd_order, _with_linear_symmetries]]	✓

19512	$x^{2} y^{''} - 3 y^{'} x + 4 y = 2 x^{2}$	[[_2nd_order, _with_linear_symmetries]]	✓

19514	$x^{2} y^{''} + 3 y^{'} x + y = \frac{1}{{(1 - x)}^{2}}$	[[_2nd_order, _exact, _linear, _nonhomogeneous]]	✓

19520	$x^{2} y^{''} - (2 m - 1) x y^{'} + (m^{2} + n^{2}) y = n^{2} x^{m} \ln (x)$	[[_2nd_order, _linear, _nonhomogeneous]]	✓

19521	$x^{2} y^{''} - 3 y^{'} x + y = \frac{\ln (x) \sin (\ln (x)) + 1}{x}$	[[_2nd_order, _linear, _nonhomogeneous]]	✓

19545	$(- x^{2} + 1) y^{''} + y^{'} x - y = x {(- x^{2} + 1)}^{3 / 2}$	[[_2nd_order, _with_linear_symmetries]]	✓

19548	$(x \sin (x) + \cos (x)) y^{''} - x \cos (x) y^{'} + \cos (x) y = 0$	[[_2nd_order, _with_linear_symmetries]]	✓

19564	$x^{2} y^{''} + y^{'} x - y = x^{2} e^{x}$	[[_2nd_order, _exact, _linear, _nonhomogeneous]]	✓

2.4.16 second order change of variable on y method 2