23.2.192 problem 197

Internal problem ID [5547]
Book : Ordinary differential equations and their solutions. By George Moseley Murphy. 1960
Section : Part II. Chapter 2. THE DIFFERENTIAL EQUATION IS OF FIRST ORDER AND OF SECOND OR HIGHER DEGREE, page 278
Problem number : 197
Date solved : Tuesday, September 30, 2025 at 12:51:53 PM
CAS classification : [[_homogeneous, `class A`], _rational, _dAlembert]

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

✓ Maple. Time used: 0.057 (sec). Leaf size: 113

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

\begin{align*} y &= \sqrt {-a}\, x \\ y &= -\sqrt {-a}\, x \\ y &= 0 \\ y &= \operatorname {RootOf}\left (-\ln \left (x \right )-\int _{}^{\textit {\_Z}}\frac {\textit {\_a}^{2}+\sqrt {a \left (\textit {\_a}^{2}+a \right )}+a}{\textit {\_a} \left (\textit {\_a}^{2}+a \right )}d \textit {\_a} +c_1 \right ) x \\ y &= \operatorname {RootOf}\left (-\ln \left (x \right )+\int _{}^{\textit {\_Z}}-\frac {\textit {\_a}^{2}-\sqrt {a \left (\textit {\_a}^{2}+a \right )}+a}{\textit {\_a} \left (\textit {\_a}^{2}+a \right )}d \textit {\_a} +c_1 \right ) x \\ \end{align*}

✓ Mathematica. Time used: 9.636 (sec). Leaf size: 88

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

\begin{align*} y(x)&\to -\sqrt {e^{c_1} \left (-2 \sqrt {a} x+e^{c_1}\right )}\\ y(x)&\to \sqrt {e^{c_1} \left (-2 \sqrt {a} x+e^{c_1}\right )}\\ y(x)&\to 0\\ y(x)&\to -i \sqrt {a} x\\ y(x)&\to i \sqrt {a} x \end{align*}

✗ Sympy

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

Timed Out