23.4.96 problem 96

Internal problem ID [6398]
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 : 96
Date solved : Tuesday, September 30, 2025 at 02:56:16 PM
CAS classification : [[_2nd_order, _missing_y], [_2nd_order, _reducible, _mu_y_y1]]

\begin{align*} {y^{\prime }}^{2}+x^{2} y^{\prime \prime }&=0 \end{align*}

✓ Maple. Time used: 0.010 (sec). Leaf size: 21

ode:=diff(y(x),x)^2+x^2*diff(diff(y(x),x),x) = 0; 
dsolve(ode,y(x), singsol=all);
 

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

✓ Mathematica. Time used: 0.31 (sec). Leaf size: 47

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

\begin{align*} y(x)&\to -\frac {x}{c_1}+\frac {\log (1+c_1 x)}{c_1{}^2}+c_2\\ y(x)&\to c_2\\ y(x)&\to -\frac {x^2}{2}+c_2 \end{align*}

✓ Sympy. Time used: 0.392 (sec). Leaf size: 17

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

\[ y{\left (x \right )} = C_{1} - \frac {x}{C_{2}} + \frac {\log {\left (C_{2} x + 1 \right )}}{C_{2}^{2}} \]