23.4.91 problem 91

Internal problem ID [6393]
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 : 91
Date solved : Friday, October 03, 2025 at 02:05:37 AM
CAS classification : [[_2nd_order, _missing_y]]

\begin{align*} 2 y^{\prime }+x y^{\prime \prime }&=a \,x^{2 k} {y^{\prime }}^{k} \end{align*}

✓ Maple. Time used: 0.014 (sec). Leaf size: 46

ode:=2*diff(y(x),x)+x*diff(diff(y(x),x),x) = a*x^(2*k)*diff(y(x),x)^k; 
dsolve(ode,y(x), singsol=all);
 

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

✓ Mathematica. Time used: 60.128 (sec). Leaf size: 58

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

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

✗ Sympy

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

NotImplementedError : multiple generators [_X0, _X0**k] 
No algorithms are implemented to solve equat