60.7.57 problem 1647 (6.57)

Internal problem ID [11646]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 6, non-linear second order
Problem number : 1647 (6.57)
Date solved : Tuesday, January 28, 2025 at 06:07:18 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} y^{\prime \prime }-a \left (-y+x y^{\prime }\right )^{v}&=0 \end{align*}

✓ Solution by Maple

Time used: 0.079 (sec). Leaf size: 59

dsolve(diff(diff(y(x),x),x)-a*(x*diff(y(x),x)-y(x))^v=0,y(x), singsol=all)
 

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

✓ Solution by Mathematica

Time used: 120.277 (sec). Leaf size: 60

DSolve[-(a*(-y[x] + x*D[y[x],x])^v) + D[y[x],{x,2}] == 0,y[x],x,IncludeSingularSolutions -> True]
 

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