38.6.19 problem 19

Internal problem ID [8449]
Book : A First Course in Differential Equations with Modeling Applications by Dennis G. Zill. 12 ed. Metric version. 2024. Cengage learning.
Section : Chapter 2. First order differential equations. Section 2.3 Linear equations. Exercises 2.3 at page 63
Problem number : 19
Date solved : Tuesday, September 30, 2025 at 05:37:10 PM
CAS classification : [_linear]

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

✓ Maple. Time used: 0.001 (sec). Leaf size: 19

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

\[ y = \frac {\left (x^{2}+c_1 \right ) {\mathrm e}^{-x}}{x +1} \]

✓ Mathematica. Time used: 0.114 (sec). Leaf size: 76

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

\begin{align*} y(x)&\to \exp \left (\int _1^x-\frac {K[1]+2}{K[1]+1}dK[1]\right ) \left (\int _1^x\frac {2 \exp \left (-K[2]-\int _1^{K[2]}-\frac {K[1]+2}{K[1]+1}dK[1]\right ) K[2]}{K[2]+1}dK[2]+c_1\right ) \end{align*}

✓ Sympy. Time used: 0.207 (sec). Leaf size: 14

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

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