6.2.18 problem 18

Internal problem ID [1554]
Book : Elementary differential equations with boundary value problems. William F. Trench. Brooks/Cole 2001
Section : Chapter 2, First order equations. Linear first order. Section 2.1 Page 41
Problem number : 18
Date solved : Tuesday, September 30, 2025 at 04:36:08 AM
CAS classification : [_linear]

\begin{align*} x y^{\prime }+\left (2 x^{2}+1\right ) y&=x^{3} {\mathrm e}^{-x^{2}} \end{align*}

✓ Maple. Time used: 0.002 (sec). Leaf size: 21

ode:=x*diff(y(x),x)+(2*x^2+1)*y(x) = x^3*exp(-x^2); 
dsolve(ode,y(x), singsol=all);
 

\[ y = \frac {\left (\frac {x^{4}}{4}+c_1 \right ) {\mathrm e}^{-x^{2}}}{x} \]

✓ Mathematica. Time used: 0.049 (sec). Leaf size: 27

ode=x*D[y[x],x] +(1+2*x^2)*y[x]==x^3*Exp[-x^2]; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)&\to \frac {e^{-x^2} \left (x^4+4 c_1\right )}{4 x} \end{align*}

✓ Sympy. Time used: 0.201 (sec). Leaf size: 15

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(-x**3*exp(-x**2) + x*Derivative(y(x), x) + (2*x**2 + 1)*y(x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

\[ y{\left (x \right )} = \left (\frac {C_{1}}{x} + \frac {x^{3}}{4}\right ) e^{- x^{2}} \]