Internal problem ID [9222]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 6, non-linear second order
Problem number: 1647 (6.57).
ODE order: 2.
ODE degree: 0.
CAS Maple gives this as type [[_2nd_order, _with_linear_symmetries]]
\[ \boxed {y^{\prime \prime }-a \left (y^{\prime } x -y\right )^{v}=0} \]
✓ Solution by Maple
Time used: 0.047 (sec). Leaf size: 123
dsolve(diff(diff(y(x),x),x)-a*(x*diff(y(x),x)-y(x))^v=0,y(x), singsol=all)
\[ y \left (x \right ) = \left (\int \left (-\frac {2^{\frac {v}{-1+v}} \left (\frac {1}{-a v \,x^{2}+a \,x^{2}+c_{1}}\right )^{\frac {v}{-1+v}} a v}{2}+\frac {2^{\frac {v}{-1+v}} \left (\frac {1}{-a v \,x^{2}+a \,x^{2}+c_{1}}\right )^{\frac {v}{-1+v}} a}{2}+\frac {2^{\frac {v}{-1+v}} \left (\frac {1}{-a v \,x^{2}+a \,x^{2}+c_{1}}\right )^{\frac {v}{-1+v}} c_{1}}{2 x^{2}}\right )d x +c_{2} \right ) x \]
✓ Solution by Mathematica
Time used: 60.421 (sec). Leaf size: 60
DSolve[-(a*(-y[x] + x*y'[x])^v) + 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 v}-\frac {1}{2} a v K[2]^{2 v}+c_1 K[2]^{2 v-2}\right ){}^{\frac {1}{1-v}}dK[2]+c_2\right ) \\ \end{align*}