Internal problem ID [9226]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 6, non-linear second order
Problem number: 1647.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _with_linear_symmetries]]
Solve \begin {gather*} \boxed {y^{\prime \prime }-a \left (y^{\prime } x -y\right )^{r}=0} \end {gather*}
✓ Solution by Maple
Time used: 0.029 (sec). Leaf size: 123
dsolve(diff(diff(y(x),x),x)-a*(x*diff(y(x),x)-y(x))^r=0,y(x), singsol=all)
\[ y \relax (x ) = \left (\int \left (-\frac {2^{\frac {r}{r -1}} \left (\frac {1}{-a r \,x^{2}+a \,x^{2}+c_{1}}\right )^{\frac {r}{r -1}} a r}{2}+\frac {2^{\frac {r}{r -1}} \left (\frac {1}{-a r \,x^{2}+a \,x^{2}+c_{1}}\right )^{\frac {r}{r -1}} a}{2}+\frac {2^{\frac {r}{r -1}} \left (\frac {1}{-a r \,x^{2}+a \,x^{2}+c_{1}}\right )^{\frac {r}{r -1}} c_{1}}{2 x^{2}}\right )d x +c_{2}\right ) x \]
✓ Solution by Mathematica
Time used: 0.347 (sec). Leaf size: 60
DSolve[-(a*(-y[x] + x*y'[x])^r) + y''[x] == 0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} 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 ) \\ \end{align*}