23.2.64 problem 66

Internal problem ID [5419]
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 : 66
Date solved : Tuesday, September 30, 2025 at 12:40:59 PM
CAS classification : [_quadrature]

\begin{align*} {y^{\prime }}^{2}-2 \left (x -y\right ) y^{\prime }-4 x y&=0 \end{align*}

✓ Maple. Time used: 0.002 (sec). Leaf size: 18

ode:=diff(y(x),x)^2-2*(x-y(x))*diff(y(x),x)-4*x*y(x) = 0; 
dsolve(ode,y(x), singsol=all);
 

\begin{align*} y &= x^{2}+c_1 \\ y &= c_1 \,{\mathrm e}^{-2 x} \\ \end{align*}

✓ Mathematica. Time used: 0.024 (sec). Leaf size: 28

ode=(D[y[x],x])^2-2*(x-y[x])*D[y[x],x]-4*x*y[x]==0; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

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

✓ Sympy. Time used: 0.099 (sec). Leaf size: 15

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(-4*x*y(x) - (2*x - 2*y(x))*Derivative(y(x), x) + Derivative(y(x), x)**2,0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

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