12.3.11 problem 12

Internal problem ID [1588]
Book : Elementary differential equations with boundary value problems. William F. Trench. Brooks/Cole 2001
Section : Chapter 2, First order equations. separable equations. Section 2.2 Page 52
Problem number : 12
Date solved : Saturday, March 29, 2025 at 11:00:48 PM
CAS classification : [_separable]

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

With initial conditions

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

✓ Maple. Time used: 0.067 (sec). Leaf size: 19

ode:=diff(y(x),x)+x*(y(x)^2+y(x)) = 0; 
ic:=y(2) = 1; 
dsolve([ode,ic],y(x), singsol=all);
 

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

✓ Mathematica. Time used: 0.217 (sec). Leaf size: 27

ode=D[y[x],x]+x*(y[x]^2+y[x])==0; 
ic=y[2]==1; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\[ y(x)\to -\frac {e^2}{e^2-2 e^{\frac {x^2}{2}}} \]

✓ Sympy. Time used: 1.646 (sec). Leaf size: 31

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

\[ y{\left (x \right )} = \frac {- \frac {\sqrt {e^{x^{2} + 4}}}{2} - \frac {e^{4}}{4}}{- e^{x^{2}} + \frac {e^{4}}{4}} \]