14.7.2 problem Problem 26

Internal problem ID [3717]
Book : Differential equations and linear algebra, Stephen W. Goode and Scott A Annin. Fourth edition, 2015
Section : Chapter 8, Linear differential equations of order n. Section 8.3, The Method of Undetermined Coefficients. page 525
Problem number : Problem 26
Date solved : Tuesday, September 30, 2025 at 06:56:18 AM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} y^{\prime \prime }+4 y^{\prime }+4 y&=5 x \,{\mathrm e}^{-2 x} \end{align*}

✓ Maple. Time used: 0.004 (sec). Leaf size: 19

ode:=diff(diff(y(x),x),x)+4*diff(y(x),x)+4*y(x) = 5*x*exp(-2*x); 
dsolve(ode,y(x), singsol=all);
 

\[ y = {\mathrm e}^{-2 x} \left (c_2 +c_1 x +\frac {5}{6} x^{3}\right ) \]

✓ Mathematica. Time used: 0.016 (sec). Leaf size: 29

ode=D[y[x],{x,2}]+4*D[y[x],x]+4*y[x]==5*x*Exp[-2*x]; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)&\to \frac {1}{6} e^{-2 x} \left (5 x^3+6 c_2 x+6 c_1\right ) \end{align*}

✓ Sympy. Time used: 0.175 (sec). Leaf size: 19

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(-5*x*exp(-2*x) + 4*y(x) + 4*Derivative(y(x), x) + Derivative(y(x), (x, 2)),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

\[ y{\left (x \right )} = \left (C_{1} + x \left (C_{2} + \frac {5 x^{2}}{6}\right )\right ) e^{- 2 x} \]