23.2.71 problem 73

Internal problem ID [5426]
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 : 73
Date solved : Tuesday, September 30, 2025 at 12:42:03 PM
CAS classification : [[_1st_order, `_with_symmetry_[F(x),G(y)]`]]

\begin{align*} {y^{\prime }}^{2}-x y^{\prime } y+y^{2} \ln \left (a y\right )&=0 \end{align*}

✓ Maple. Time used: 0.080 (sec). Leaf size: 43

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

\begin{align*} y &= \frac {{\mathrm e}^{\frac {x^{2}}{4}}}{a} \\ y &= \frac {{\mathrm e}^{c_1 \left (-c_1 +x \right )}}{a} \\ y &= \frac {{\mathrm e}^{-c_1 \left (c_1 +x \right )}}{a} \\ \end{align*}

✓ Mathematica. Time used: 0.182 (sec). Leaf size: 30

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

\begin{align*} y(x)&\to \frac {e^{\frac {1}{4} c_1 (2 x-c_1)}}{a}\\ y(x)&\to 0 \end{align*}

✗ Sympy

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

Timed Out