second order change of variable on y method 1

Table 2.481: second order change of variable on y method 1


#	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)]‘]]	✓

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

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

526	$x y^{''} + 2 y^{'} + x y = 0$	[_Lienard]	✓

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

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

1294	$t^{2} y^{''} + 4 y^{'} t + 2 y = 0$	[[_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)]‘]]	✓

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

1350	$x^{2} y^{''} + y^{'} x + (x^{2} - \frac{1}{4}) y = g (x)$	[[_2nd_order, _linear, _nonhomogeneous]]	✓

1742	$(x^{2} - 1) y^{''} + 4 y^{'} x + 2 y = 0$ i.c.	[[_2nd_order, _exact, _linear, _homogeneous]]	✓

1749	$4 x^{2} y^{''} - 4 y^{'} x + (- 16 x^{2} + 3) y = 0$	[[_2nd_order, _with_linear_symmetries]]	✓

1751	$x^{2} y^{''} - 2 y^{'} x + (x^{2} + 2) y = 0$	[[_2nd_order, _with_linear_symmetries]]	✓

1754	$(x^{2} - 4) y^{''} + 4 y^{'} x + 2 y = 0$	[[_2nd_order, _exact, _linear, _homogeneous]]	✓

1812	$x y^{''} + (2 - 2 x) y^{'} + (x - 2) y = e^{2 x}$	[[_2nd_order, _linear, _nonhomogeneous]]	✓

1813	$4 x^{2} y^{''} + (- 8 x^{2} + 4 x) y^{'} + (4 x^{2} - 4 x - 1) y = 4 \sqrt{x} e^{x}$	[[_2nd_order, _linear, _nonhomogeneous]]	✓

1814	$y^{''} + 4 y^{'} x + (4 x^{2} + 2) y = 4 e^{- x (x + 2)}$	[[_2nd_order, _linear, _nonhomogeneous]]	✓

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

1821	$x^{2} y^{''} - 2 y^{'} x + (x^{2} + 2) y = x^{3} \cos (x)$	[[_2nd_order, _linear, _nonhomogeneous]]	✓

1824	$4 x^{2} y^{''} - 4 y^{'} x + (- 16 x^{2} + 3) y = 8 x^{5 / 2}$	[[_2nd_order, _linear, _nonhomogeneous]]	✓

1825	$4 x^{2} y^{''} - 4 y^{'} x + (4 x^{2} + 3) y = x^{7 / 2}$	[[_2nd_order, _linear, _nonhomogeneous]]	✓

1826	$x^{2} y^{''} - 2 y^{'} x - (x^{2} - 2) y = 3 x^{4}$	[[_2nd_order, _linear, _nonhomogeneous]]	✓

1827	$x^{2} y^{''} - 2 x (x + 1) y^{'} + (x^{2} + 2 x + 2) y = x^{3} e^{x}$	[[_2nd_order, _linear, _nonhomogeneous]]	✓

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

1830	$x^{2} y^{''} - 2 x (x + 2) y^{'} + (x^{2} + 4 x + 6) y = 2 x e^{x}$	[[_2nd_order, _linear, _nonhomogeneous]]	✓

1831	$x^{2} y^{''} - 4 y^{'} x + (x^{2} + 6) y = x^{4}$	[[_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]]	✓

1836	${(x - 1)}^{2} y^{''} - (x^{2} - 1) y^{'} + (x + 1) y = {(x - 1)}^{3} e^{x}$ i.c.	[[_2nd_order, _with_linear_symmetries]]	✓

2394	$y^{''} - 4 y^{'} t + (4 t^{2} - 2) y = 0$	[[_2nd_order, _with_linear_symmetries]]	✓

2399	$t^{2} y^{''} + y^{'} t + (t^{2} - \frac{1}{4}) y = 0$	[[_2nd_order, _with_linear_symmetries]]	✓

2434	${(t - 1)}^{2} y^{''} - 2 (t - 1) y^{'} + 2 y = 0$	[[_2nd_order, _with_linear_symmetries], [_2nd_order, _linear, ‘_with_symmetry_[0,F(x)]‘]]	✓

2630	${(t - 1)}^{2} y^{''} - 2 (t - 1) y^{'} + 2 y = 0$	[[_2nd_order, _with_linear_symmetries], [_2nd_order, _linear, ‘_with_symmetry_[0,F(x)]‘]]	✓

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

3500	$y^{''} + 4 y^{'} x + (4 x^{2} + 6) y = e^{- x^{2}} \sin (2 x)$	[[_2nd_order, _linear, _nonhomogeneous]]	✓

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

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]]	✓

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]]	✓

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

5991	$y^{''} - \frac{2 y^{'}}{x} + \frac{2 y}{x^{2}} = x \ln (x)$	[[_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)]‘]]	✓

6078	$u^{''} + \frac{2 u^{'}}{x} - a^{2} u = 0$	[[_2nd_order, _with_linear_symmetries]]	✓

6079	$u^{''} + \frac{2 u^{'}}{x} + a^{2} u = 0$	[[_2nd_order, _with_linear_symmetries]]	✓

6084	$x^{2} y^{''} + y^{'} x - (x^{2} + \frac{1}{4}) y = 0$	[[_2nd_order, _with_linear_symmetries]]	✓

6255	$y^{''} - 4 y^{'} x + (4 x^{2} - 2) y = 0$	[[_2nd_order, _with_linear_symmetries]]	✓

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

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

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

6759	$y^{''} - 2 y^{'} \tan (x) - 10 y = 0$	[[_2nd_order, _with_linear_symmetries]]	✓

6769	$x^{2} y^{''} - 4 y^{'} x + (9 x^{2} + 6) y = 0$	[[_2nd_order, _with_linear_symmetries]]	✓

6770	$x y^{''} + 2 y^{'} + 4 x y = 4$	[[_2nd_order, _linear, _nonhomogeneous]]	✓

7498	$x y^{''} + 2 y^{'} + x y = \sec (x)$	[[_2nd_order, _linear, _nonhomogeneous]]	✓

7523	$y^{''} + 2 x^{2} y^{'} + (x^{4} + 2 x - 1) y = 0$	[[_2nd_order, _with_linear_symmetries]]	✓

7525	$\sin (x) u^{''} + 2 \cos (x) u^{'} + \sin (x) u = 0$	[_Lienard]	✓

7535	$u^{''} - (2 x + 1) u^{'} + (x^{2} + x - 1) u = 0$	[[_2nd_order, _with_linear_symmetries]]	✓

7545	$x^{2} y^{''} - 2 y^{'} x + (x^{2} + 2) y = 0$	[[_2nd_order, _with_linear_symmetries]]	✓

7686	$x^{2} y^{''} + 4 y^{'} x + (x^{2} + 2) y = 0$	[[_2nd_order, _with_linear_symmetries]]	✓

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

8067	$x^{2} y^{''} - 2 y^{'} x + 2 y = 0$ i.c.	[[_Emden, _Fowler], [_2nd_order, _linear, ‘_with_symmetry_[0,F(x)]‘]]	✓

8294	$x^{2} y^{''} + y^{'} x + (36 x^{2} - \frac{1}{4}) y = 0$	[[_2nd_order, _with_linear_symmetries]]	✓

8308	$x^{2} y^{''} + 4 y^{'} x + (x^{2} + 2) y = 0$	[[_2nd_order, _with_linear_symmetries]]	✓

8310	$4 x^{2} y^{''} - 4 y^{'} x + (16 x^{2} + 3) y = 0$	[[_2nd_order, _with_linear_symmetries]]	✓

8963	$y^{''} + 2 y^{'} x + (x^{2} + 1) y = 0$	[[_2nd_order, _with_linear_symmetries]]	✓

8964	$y^{''} + 2 \cot (x) y^{'} - y = 0$	[[_2nd_order, _with_linear_symmetries]]	✓

8965	$x^{2} y^{''} + y^{'} x + (x^{2} - \frac{1}{4}) y = 0$	[[_2nd_order, _with_linear_symmetries]]	✓

8966	$4 x^{2} y^{''} + (- 8 x^{2} + 4 x) y^{'} + (4 x^{2} - 4 x - 1) y = 4 \sqrt{x} e^{x}$	[[_2nd_order, _linear, _nonhomogeneous]]	✓

9152	$\cos {(x)}^{2} y^{''} - 2 \cos (x) \sin (x) y^{'} + \cos {(x)}^{2} y = 0$	[_Lienard]	✓

9153	$y^{''} - 4 y^{'} x + (4 x^{2} - 1) y = - 3 e^{x^{2}} \sin (x)$	[[_2nd_order, _linear, _nonhomogeneous]]	✓

9154	$y^{''} - 2 b x y^{'} + b^{2} x^{2} y = x$	[[_2nd_order, _linear, _nonhomogeneous]]	✓

9155	$y^{''} - 4 y^{'} x + (4 x^{2} - 3) y = e^{x^{2}}$	[[_2nd_order, _linear, _nonhomogeneous]]	✓

9156	$y^{''} - 2 y^{'} \tan (x) + 5 y = e^{x^{2}} \sec (x)$	[[_2nd_order, _linear, _nonhomogeneous]]	✓

9157	$x^{2} y^{''} - 2 y^{'} x + 2 (x^{2} + 1) y = 0$	[[_2nd_order, _with_linear_symmetries]]	✓

9160	$x y^{''} + 2 y^{'} - x y = 0$	[[_2nd_order, _with_linear_symmetries]]	✓

9161	$x y^{''} + 2 y^{'} + x y = 0$	[_Lienard]	✓

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

11038	$y^{''} + 4 y^{'} x + (4 x^{2} + 2) y = 0$	[[_2nd_order, _with_linear_symmetries]]	✓

11041	$y^{''} - 4 y^{'} x + (4 x^{2} - 2) y = 0$	[[_2nd_order, _with_linear_symmetries]]	✓

11042	$y^{''} - 4 y^{'} x + (4 x^{2} - 3) y - e^{x^{2}} = 0$	[[_2nd_order, _linear, _nonhomogeneous]]	✓

11044	$y^{''} + 2 a x y^{'} + a^{2} x^{2} y = 0$	[[_2nd_order, _with_linear_symmetries]]	✓

11052	$y^{''} + \sqrt{x} y^{'} + (\frac{1}{4 \sqrt{x}} + \frac{x}{4} - 9) y - x e^{- \frac{x^{3 / 2}}{3}} = 0$	[[_2nd_order, _linear, _nonhomogeneous]]	✓

11063	$y^{''} + f (x) y^{'} + (\frac{f {(x)}^{2}}{4} + \frac{f^{'} (x)}{2} + a) y = 0$	[[_2nd_order, _with_linear_symmetries]]	✓

11072	$a^{2} y^{''} + a (a^{2} - 2 b e^{- a x}) y^{'} + b^{2} e^{- 2 a x} y = 0$	[[_2nd_order, _with_linear_symmetries]]	✓

11082	$x y^{''} + 2 y^{'} - x y - e^{x} = 0$	[[_2nd_order, _linear, _nonhomogeneous]]	✓

11083	$x y^{''} + 2 y^{'} + y a x = 0$	[[_2nd_order, _with_linear_symmetries]]	✓

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

11158	$x^{2} y^{''} - 2 y^{'} x + (x^{2} + 2) y = 0$	[[_2nd_order, _with_linear_symmetries]]	✓

11160	$x^{2} y^{''} - 2 y^{'} x + (x^{2} + 2) y - \frac{x^{3}}{\cos (x)} = 0$	[[_2nd_order, _linear, _nonhomogeneous]]	✓

11161	$x^{2} y^{''} - 2 y^{'} x + (a^{2} x^{2} + 2) y = 0$	[[_2nd_order, _with_linear_symmetries]]	✓

11184	$x^{2} y^{''} - 2 x (x + 1) y^{'} + 2 (x + 1) y = 0$	[[_2nd_order, _with_linear_symmetries]]	✓

11210	$(x^{2} + 1) y^{''} + 4 y^{'} x + 2 y - 2 \cos (x) + 2 x = 0$	[[_2nd_order, _exact, _linear, _nonhomogeneous]]	✓

11223	$(x^{2} - 1) y^{''} + 4 y^{'} x + (x^{2} + 1) y = 0$	[[_2nd_order, _with_linear_symmetries]]	✓

11257	$4 x^{2} y^{''} + 4 y^{'} x - (a x^{2} + 1) y = 0$	[[_2nd_order, _with_linear_symmetries]]	✓

11261	$4 x^{2} y^{''} - 4 x (2 x - 1) y^{'} + (4 x^{2} - 4 x - 1) y = 0$	[[_2nd_order, _with_linear_symmetries]]	✓

12467	$y^{''} + 2 a x^{n} y^{'} + a (a x^{2 n} + n x^{n - 1}) y = 0$	[[_2nd_order, _with_linear_symmetries]]	✓

12550	$x^{2} y^{''} - 2 a x y^{'} + (b^{2} x^{2} + a (a + 1)) y = 0$	[[_2nd_order, _with_linear_symmetries]]	✓

12551	$x^{2} y^{''} - 2 a x y^{'} + (- b^{2} x^{2} + a (a + 1)) y = 0$	[[_2nd_order, _with_linear_symmetries]]	✓

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]]	✓

12698	$y^{''} + 2 a e^{λ x} y^{'} + a e^{λ x} (a e^{λ x} + λ) y = 0$	[[_2nd_order, _with_linear_symmetries]]	✓

12702	$y^{''} - (a + 2 b e^{a x}) y^{'} + b^{2} e^{2 a x} y = 0$	[[_2nd_order, _with_linear_symmetries]]	✓

12898	$\sin (x) y^{''} + 2 \cos (x) y^{'} + 3 y \sin (x) = e^{x}$	[[_2nd_order, _linear, _nonhomogeneous]]	✓

12899	$y^{''} - 2 y^{'} \tan (x) - (a^{2} + 1) y = 0$	[[_2nd_order, _with_linear_symmetries]]	✓

12901	$x y^{''} + 2 y^{'} - x y = 2 e^{x}$	[[_2nd_order, _linear, _nonhomogeneous]]	✓

12902	$y^{''} + (2 e^{x} - 1) y^{'} + e^{2 x} y = e^{4 x}$	[[_2nd_order, _linear, _nonhomogeneous]]	✓

12909	$x^{2} y^{''} + 4 y^{'} x + (- x^{2} + 2) y = 0$	[[_2nd_order, _with_linear_symmetries]]	✓

12912	$x^{2} y^{''} - 4 y^{'} x + (x^{2} + 6) y = 0$	[[_2nd_order, _with_linear_symmetries]]	✓

12914	$x^{2} y^{''} - 2 x (x + 1) y^{'} + 2 (x + 1) y = x^{3}$	[[_2nd_order, _with_linear_symmetries]]	✓

12930	${(x - 1)}^{2} y^{''} + 4 (x - 1) y^{'} + 2 y = \cos (x)$	[[_2nd_order, _exact, _linear, _nonhomogeneous]]	✓

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

13178	$(x^{2} + 1) y^{''} + 4 y^{'} x + 2 y = 0$	[[_2nd_order, _exact, _linear, _homogeneous]]	✓

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

13453	${(x + 1)}^{2} y^{''} - 2 (x + 1) y^{'} + 2 y = 1$	[[_2nd_order, _with_linear_symmetries]]	✓

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

13458	$\sin {(x)}^{2} y^{''} - 2 \cos (x) \sin (x) y^{'} + (\cos {(x)}^{2} + 1) y = \sin {(x)}^{3}$	[[_2nd_order, _linear, _nonhomogeneous]]	✓

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

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

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

13480	$x^{2} y^{''} - 4 y^{'} x + 6 y = 0$ 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)]‘]]	✓

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

13926	$y^{''} + 4 y^{'} x + (4 x^{2} + 2) y = 0$	[[_2nd_order, _with_linear_symmetries]]	✓

13933	$x \ln (x) y^{''} + 2 y^{'} - \frac{y}{x} = 1$	[[_2nd_order, _exact, _linear, _nonhomogeneous]]	✓

14063	$y^{''} + \frac{y^{'}}{x} + (1 - \frac{1}{4 x^{2}}) y = x$	[[_2nd_order, _linear, _nonhomogeneous]]	✓

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)]‘]]	✓

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)]‘]]	✓

15233	${(x + 1)}^{2} y^{''} - 2 (x + 1) y^{'} + 2 y = 0$ i.c.	[[_2nd_order, _with_linear_symmetries], [_2nd_order, _linear, ‘_with_symmetry_[0,F(x)]‘]]	✓

15234	$x^{2} y^{''} - 4 y^{'} x + 6 y = 0$	[[_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)]‘]]	✓

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]]	✓

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

15454	$x y^{''} + (2 x + 2) y^{'} + 2 y = 8 e^{2 x}$	[[_2nd_order, _exact, _linear, _nonhomogeneous]]	✓

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

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

16289	$t^{2} y^{''} - 4 y^{'} t + (t^{2} + 6) y = t^{3} + 2 t$ i.c.	[[_2nd_order, _with_linear_symmetries]]	✓

16291	$t y^{''} + 2 y^{'} + t y = - t$ i.c.	[[_2nd_order, _with_linear_symmetries]]	✓

16293	$4 t^{2} y^{''} + 4 y^{'} t + (16 t^{2} - 1) y = 16 t^{3 / 2}$ i.c.	[[_2nd_order, _linear, _nonhomogeneous]]	✓

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

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

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

16532	$y^{''} - 2 y^{'} t + t^{2} y = 0$	[[_2nd_order, _with_linear_symmetries]]	✓

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

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

17146	$x^{2} y^{''} + y^{'} x + (x^{2} - \frac{1}{4}) y = 0$	[[_2nd_order, _with_linear_symmetries]]	✓

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

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

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

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

17631	$x^{2} y^{''} + y^{'} x + (x^{2} - \frac{1}{4}) y = 3 x^{3 / 2} \sin (x)$	[[_2nd_order, _linear, _nonhomogeneous]]	✓

17633	$x^{2} y^{''} + y^{'} x + (x^{2} - \frac{1}{4}) y = g (x)$	[[_2nd_order, _linear, _nonhomogeneous]]	✓

17925	$y^{''} + \frac{2 y^{'}}{x} + y = 0$	[_Lienard]	✓

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

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

17962	$y^{''} + \frac{2 y^{'}}{x} + y = 0$	[_Lienard]	✓

17965	$y^{''} - 4 y^{'} x + (4 x^{2} - 1) y = - 3 e^{x^{2}} \sin (2 x)$	[[_2nd_order, _linear, _nonhomogeneous]]	✓

18190	$x^{2} y^{''} - 4 y^{'} x + (x^{2} + 6) y = 0$	[[_2nd_order, _with_linear_symmetries]]	✓

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

18200	$y^{''} + 2 y^{'} x + (x^{2} + 1) y = 0$	[[_2nd_order, _with_linear_symmetries]]	✓

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

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

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

18531	$x y^{''} + 2 y^{'} = x y$	[[_2nd_order, _with_linear_symmetries]]	✓

18547	$v^{''} + \frac{2 x v^{'}}{x^{2} + 1} + \frac{v}{{(x^{2} + 1)}^{2}} = 0$	[[_2nd_order, _with_linear_symmetries], [_2nd_order, _linear, ‘_with_symmetry_[0,F(x)]‘]]	✓

18620	$(x^{2} - 1) y^{''} + 4 y^{'} x + 2 y = 2 x$	[[_2nd_order, _exact, _linear, _nonhomogeneous]]	✓

18621	$(x^{2} + 1) y^{''} + 4 y^{'} x + 2 y = x$	[[_2nd_order, _exact, _linear, _nonhomogeneous]]	✓

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]]	✓

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

18865	${(x + a)}^{2} y^{''} - 4 (x + a) y^{'} + 6 y = x$	[[_2nd_order, _with_linear_symmetries]]	✓

18940	$y^{''} - 2 y^{'} \tan (x) + 5 y = 0$	[[_2nd_order, _with_linear_symmetries]]	✓

18941	$x^{2} y^{''} - 2 (x^{2} + x) y^{'} + (x^{2} + 2 x + 2) y = 0$	[[_2nd_order, _with_linear_symmetries]]	✓

18946	$y^{''} + \frac{2 y^{'}}{x} = n^{2} y$	[[_2nd_order, _with_linear_symmetries]]	✓

18947	$y^{''} + \frac{2 y^{'}}{x} + n^{2} y = 0$	[[_2nd_order, _with_linear_symmetries]]	✓

18948	$y^{''} - \frac{2 y^{'}}{x} + (n^{2} + \frac{2}{x^{2}}) y = 0$	[[_2nd_order, _with_linear_symmetries]]	✓

18952	$y^{''} + 4 y^{'} x + 4 x^{2} y = 0$	[[_2nd_order, _with_linear_symmetries]]	✓

18959	$x^{2} y^{''} - 2 x (x + 1) y^{'} + 2 (x + 1) y = x^{3}$	[[_2nd_order, _with_linear_symmetries]]	✓

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

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

19375	$y^{''} + 4 y^{'} x + 4 x^{2} y = 0$	[[_2nd_order, _with_linear_symmetries]]	✓

19376	$y^{''} + \frac{2 y^{'}}{x} + n^{2} y = 0$	[[_2nd_order, _with_linear_symmetries]]	✓

19377	$y^{''} + \frac{2 y^{'}}{x} = n^{2} y$	[[_2nd_order, _with_linear_symmetries]]	✓

19378	$y^{''} - 2 b x y^{'} + b^{2} x^{2} y = 0$	[[_2nd_order, _with_linear_symmetries]]	✓

19379	$y^{''} - 2 b x y^{'} + b^{2} x^{2} y = x$	[[_2nd_order, _linear, _nonhomogeneous]]	✓

19382	$y^{''} - 2 y^{'} \tan (x) + 5 y = e^{x} \sec (x)$	[[_2nd_order, _linear, _nonhomogeneous]]	✓

19383	$y^{''} - 2 y^{'} \tan (x) - (a^{2} + 1) y = 0$	[[_2nd_order, _with_linear_symmetries]]	✓

19384	$y^{''} - \frac{2 y^{'}}{x} + (n^{2} + \frac{2}{x^{2}}) y = 0$	[[_2nd_order, _with_linear_symmetries]]	✓

19385	$y^{''} + 2 n \cot (n x) y^{'} + (m^{2} - n^{2}) y = 0$	[[_2nd_order, _with_linear_symmetries]]	✓

19387	$x^{2} y^{''} - 2 n x y^{'} + (a^{2} x^{2} + n^{2} + n) y = 0$	[[_2nd_order, _with_linear_symmetries]]	✓

19388	$y^{''} - 4 y^{'} x + (4 x^{2} - 3) y = e^{x^{2}}$	[[_2nd_order, _linear, _nonhomogeneous]]	✓

19407	$(- x^{2} + 1) y^{''} - 4 y^{'} x - (x^{2} + 1) y = x$	[[_2nd_order, _linear, _nonhomogeneous]]	✓

19408	$x^{2} y^{''} - 2 x (x + 1) y^{'} + 2 (x + 1) y = - 4 x^{3}$	[[_2nd_order, _with_linear_symmetries]]	✓

19411	$x^{2} y^{''} - 2 x (x + 1) y^{'} + 2 (x + 1) y = x^{3}$	[[_2nd_order, _with_linear_symmetries]]	✓


19413	$y^{''} - \frac{2 y^{'}}{x} + (n^{2} + \frac{2}{x^{2}}) y = 0$	[[_2nd_order, _with_linear_symmetries]]	✓

19420	$y^{''} + 2 y^{'} x + (x^{2} + 5) y = x e^{- \frac{x^{2}}{2}}$	[[_2nd_order, _linear, _nonhomogeneous]]	✓

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

19516	${(x + a)}^{2} y^{''} - 4 (x + a) y^{'} + 6 y = x$	[[_2nd_order, _with_linear_symmetries]]	✓

19550	$x^{2} y^{''} - 2 (x^{2} + x) y^{'} + (x^{2} + 2 x + 2) y = 0$	[[_2nd_order, _with_linear_symmetries]]	✓

19551	$x^{2} y^{''} - 2 y^{'} x + (x^{2} + 2) y = 0$	[[_2nd_order, _with_linear_symmetries]]	✓

19553	$y^{''} - 2 y^{'} \tan (x) + y = 0$	[_Lienard]	✓

19554	$y^{''} - 4 y^{'} x + (4 x^{2} - 1) y = - 3 e^{x^{2}} \sin (2 x)$	[[_2nd_order, _linear, _nonhomogeneous]]	✓

19565	$x^{2} y^{''} - 2 x (x + 1) y^{'} + 2 (x + 1) y = x^{3}$	[[_2nd_order, _with_linear_symmetries]]	✓

2.4.15 second order change of variable on y method 1