23.2.59 problem 61

Internal problem ID [5414]
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 : 61
Date solved : Tuesday, September 30, 2025 at 12:40:01 PM
CAS classification : [_quadrature]

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

✓ Maple. Time used: 0.002 (sec). Leaf size: 25

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

\begin{align*} y &= \frac {x^{2}}{2}+c_1 \\ y &= -x +1+{\mathrm e}^{-x} c_1 \\ \end{align*}

✓ Mathematica. Time used: 0.03 (sec). Leaf size: 32

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

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

✓ Sympy. Time used: 0.115 (sec). Leaf size: 19

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

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