60.7.45 problem 1647 (6.57)

Internal problem ID [11595]
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 : Sunday, March 30, 2025 at 08:30:41 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

✓ Maple. Time used: 0.040 (sec). Leaf size: 59

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

\[ y = \left (2^{\frac {1}{v -1}} \int -\frac {\left (-\frac {1}{a \left (v -1\right ) x^{2}-c_1}\right )^{\frac {v}{v -1}} \left (a \left (v -1\right ) x^{2}-c_1 \right )}{x^{2}}d x +c_2 \right ) x \]

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

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

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

✗ Sympy

from sympy import * 
x = symbols("x") 
a = symbols("a") 
v = symbols("v") 
y = Function("y") 
ode = Eq(-a*(x*Derivative(y(x), x) - y(x))**v + Derivative(y(x), (x, 2)),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

NotImplementedError : The given ODE Derivative(y(x), x) - ((Derivative(y(x), (x, 2))/a)**(1/v) + y(x))/x cannot be solved by the factorable group method