2.11.14 problem 25

Internal problem ID [882]
Book : Differential equations and linear algebra, 3rd ed., Edwards and Penney
Section : Section 5.5, Nonhomogeneous equations and undetermined coefficients Page 351
Problem number : 25
Date solved : Tuesday, September 30, 2025 at 04:18:59 AM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

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

✓ Maple. Time used: 0.003 (sec). Leaf size: 36

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

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

✓ Mathematica. Time used: 0.023 (sec). Leaf size: 42

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

\begin{align*} y(x)&\to \frac {1}{2} e^{-2 x} \left (x^2+e^x \left (x^2-2 x+2+2 c_2\right )+2 x+2+2 c_1\right ) \end{align*}

✓ Sympy. Time used: 0.260 (sec). Leaf size: 26

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

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