57.9.4 problem 1(d)

Internal problem ID [14412]
Book : A First Course in Differential Equations by J. David Logan. Third Edition. Springer-Verlag, NY. 2015.
Section : Chapter 2, Second order linear equations. Section 2.3.1 Nonhomogeneous Equations: Undetermined Coefficients. Exercises page 110
Problem number : 1(d)
Date solved : Thursday, October 02, 2025 at 09:36:52 AM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

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

✓ Maple. Time used: 0.005 (sec). Leaf size: 47

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

\[ x = {\mathrm e}^{-\frac {t}{2}} \sin \left (\frac {\sqrt {3}\, t}{2}\right ) c_2 +{\mathrm e}^{-\frac {t}{2}} \cos \left (\frac {\sqrt {3}\, t}{2}\right ) c_1 +\frac {\left (169 t^{2}-182 t +72\right ) {\mathrm e}^{3 t}}{2197} \]

✓ Mathematica. Time used: 0.012 (sec). Leaf size: 62

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

\begin{align*} x(t)&\to e^{-t/2} \left (3 \exp e^{t/2} \left (t^3-3 t^2+6\right )+c_2 \cos \left (\frac {\sqrt {3} t}{2}\right )+c_1 \sin \left (\frac {\sqrt {3} t}{2}\right )\right ) \end{align*}

✓ Sympy. Time used: 0.175 (sec). Leaf size: 48

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

\[ x{\left (t \right )} = \left (C_{1} \sin {\left (\frac {\sqrt {3} t}{2} \right )} + C_{2} \cos {\left (\frac {\sqrt {3} t}{2} \right )}\right ) e^{- \frac {t}{2}} + \frac {\left (169 t^{2} - 182 t + 72\right ) e^{3 t}}{2197} \]