23.4.58 problem 58

Internal problem ID [6360]
Book : Ordinary differential equations and their solutions. By George Moseley Murphy. 1960
Section : Part II. Chapter 4. THE NONLINEAR EQUATION OF SECOND ORDER, page 380
Problem number : 58
Date solved : Friday, October 03, 2025 at 02:05:22 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

✓ Maple. Time used: 0.049 (sec). Leaf size: 59

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

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

✓ Mathematica. Time used: 120.261 (sec). Leaf size: 60

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

\begin{align*} y(x)&\to x \left (\int _1^x\left (\frac {1}{2} a K[2]^{2 k}-\frac {1}{2} a k K[2]^{2 k}+c_1 K[2]^{2 k-2}\right ){}^{\frac {1}{1-k}}dK[2]+c_2\right ) \end{align*}

✗ Sympy

from sympy import * 
x = symbols("x") 
a = symbols("a") 
k = symbols("k") 
y = Function("y") 
ode = Eq(-a*(x*Derivative(y(x), x) - y(x))**k + 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/k) + y(x