87.4.19 problem 30

Internal problem ID [23285]
Book : Ordinary differential equations with modern applications. Ladas, G. E. and Finizio, N. Wadsworth Publishing. California. 1978. ISBN 0-534-00552-7. QA372.F56
Section : Chapter 1. Elementary methods. First order differential equations. Exercise at page 37
Problem number : 30
Date solved : Thursday, October 02, 2025 at 09:28:34 PM
CAS classification : [_linear]

\begin{align*} y^{\prime }-y x&=\left (-x^{2}+1\right ) {\mathrm e}^{\frac {x^{2}}{2}} \end{align*}

With initial conditions

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

✓ Maple. Time used: 0.012 (sec). Leaf size: 18

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

\[ y = -\frac {x \left (x^{2}-3\right ) {\mathrm e}^{\frac {x^{2}}{2}}}{3} \]

✓ Mathematica. Time used: 0.037 (sec). Leaf size: 24

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

\begin{align*} y(x)&\to -\frac {1}{3} e^{\frac {x^2}{2}} x \left (x^2-3\right ) \end{align*}

✓ Sympy. Time used: 0.198 (sec). Leaf size: 15

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

\[ y{\left (x \right )} = \left (- \frac {x^{3}}{3} + x\right ) e^{\frac {x^{2}}{2}} \]