41.2.23 problem 23

Internal problem ID [8725]
Book : Ordinary differential equations and calculus of variations. Makarets and Reshetnyak. Wold Scientific. Singapore. 1995
Section : Chapter 1. First order differential equations. Section 1.2 Homogeneous equations problems. page 12
Problem number : 23
Date solved : Tuesday, September 30, 2025 at 05:44:20 PM
CAS classification : [_quadrature]

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

With initial conditions

\begin{align*} y \left (0\right )&=0 \\ \end{align*}

✓ Maple. Time used: 0.023 (sec). Leaf size: 9

ode:=diff(y(x),x)*(diff(y(x),x)+y(x)) = x*(x+y(x)); 
ic:=[y(0) = 0]; 
dsolve([ode,op(ic)],y(x), singsol=all);
 

\[ y = \frac {x^{2}}{2} \]

✓ Mathematica. Time used: 0.03 (sec). Leaf size: 28

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

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

✓ Sympy. Time used: 0.120 (sec). Leaf size: 17

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),0) 
ics = {y(0): 0} 
dsolve(ode,func=y(x),ics=ics)
 

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