41.1.9 problem 9

Internal problem ID [8676]
Book : Ordinary differential equations and calculus of variations. Makarets and Reshetnyak. Wold Scientific. Singapore. 1995
Section : Chapter 1. First order differential equations. Section 1.1 Separable equations problems. page 7
Problem number : 9
Date solved : Tuesday, September 30, 2025 at 05:40:46 PM
CAS classification : [_separable]

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

With initial conditions

\begin{align*} y \left (1\right )&={\frac {1}{2}} \\ \end{align*}

✓ Maple. Time used: 0.027 (sec). Leaf size: 9

ode:=x*diff(y(x),x)+y(x) = y(x)^2; 
ic:=[y(1) = 1/2]; 
dsolve([ode,op(ic)],y(x), singsol=all);
 

\[ y = \frac {1}{x +1} \]

✓ Mathematica. Time used: 0.149 (sec). Leaf size: 10

ode=x*D[y[x],x]+y[x]==y[x]^2; 
ic={y[1]==1/2}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

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

✓ Sympy. Time used: 0.164 (sec). Leaf size: 10

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(x*Derivative(y(x), x) - y(x)**2 + y(x),0) 
ics = {y(1): 1/2} 
dsolve(ode,func=y(x),ics=ics)
 

\[ y{\left (x \right )} = - \frac {1}{- x - 1} \]