2.11.4 problem 4

Internal problem ID [872]
Book : Differential equations and linear algebra, 3rd ed., Edwards and Penney
Section : Section 5.5, Nonhomogeneous equations and undetermined coefficients Page 351
Problem number : 4
Date solved : Tuesday, September 30, 2025 at 04:18:51 AM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} 4 y^{\prime \prime }+4 y^{\prime }+y&=3 x \,{\mathrm e}^{x} \end{align*}

✓ Maple. Time used: 0.004 (sec). Leaf size: 24

ode:=4*diff(diff(y(x),x),x)+4*diff(y(x),x)+y(x) = 3*x*exp(x); 
dsolve(ode,y(x), singsol=all);
 

\[ y = \frac {\left (3 x -4\right ) {\mathrm e}^{x}}{9}+{\mathrm e}^{-\frac {x}{2}} \left (c_1 x +c_2 \right ) \]

✓ Mathematica. Time used: 0.012 (sec). Leaf size: 33

ode=4*D[y[x],{x,2}]+4*D[y[x],x]+y[x]==3*x*Exp[x]; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)&\to \frac {1}{9} e^x (3 x-4)+e^{-x/2} (c_2 x+c_1) \end{align*}

✓ Sympy. Time used: 0.177 (sec). Leaf size: 22

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(-3*x*exp(x) + y(x) + 4*Derivative(y(x), x) + 4*Derivative(y(x), (x, 2)),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

\[ y{\left (x \right )} = \left (C_{1} + C_{2} x\right ) e^{- \frac {x}{2}} + \frac {\left (3 x - 4\right ) e^{x}}{9} \]