15.11.7 problem 7

Internal problem ID [3117]
Book : Differential Equations by Alfred L. Nelson, Karl W. Folley, Max Coral. 3rd ed. DC heath. Boston. 1964
Section : Exercise 19, page 86
Problem number : 7
Date solved : Tuesday, March 04, 2025 at 04:00:27 PM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} y^{\prime \prime }+3 y^{\prime }+2 y&=x \,{\mathrm e}^{-x} \end{align*}

✓ Maple. Time used: 0.002 (sec). Leaf size: 27

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

\[ y = \frac {\left (-2 \,{\mathrm e}^{-x} c_{1} +x^{2}+2 c_2 -2 x \right ) {\mathrm e}^{-x}}{2} \]

✓ Mathematica. Time used: 0.027 (sec). Leaf size: 35

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

\[ y(x)\to \frac {1}{2} e^{-2 x} \left (e^x \left (x^2-2 x+2+2 c_2\right )+2 c_1\right ) \]

✓ Sympy. Time used: 0.278 (sec). Leaf size: 19

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

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