2.1647   ODE No. 1647

\[ y''(x)-a \left (x y'(x)-y(x)\right )^r=0 \]

✓ Mathematica : cpu = 0.596504 (sec), leaf count = 60

DSolve[-(a*(-y[x] + x*Derivative[1][y][x])^r) + Derivative[2][y][x] == 0,y[x],x]
 

\[\left \{\left \{y(x)\to x \left (\int _1^x\left (\frac {1}{2} a K[2]^{2 r}-\frac {1}{2} a r K[2]^{2 r}+c_1 K[2]^{2 r-2}\right ){}^{\frac {1}{1-r}}dK[2]+c_2\right )\right \}\right \}\]

✓ Maple : cpu = 8.603 (sec), leaf count = 60

dsolve(diff(diff(y(x),x),x)-a*(x*diff(y(x),x)-y(x))^r=0,y(x))
 

\[y \left (x \right ) = \left (\int -\frac {\left (-\frac {1}{\left (r -1\right ) a \,x^{2}-c_{1}}\right )^{\frac {r}{r -1}} 2^{\frac {r}{r -1}} \left (\left (r -1\right ) a \,x^{2}-c_{1} \right )}{2 x^{2}}d x +c_{2} \right ) x\]