57.9.6 problem 1(f)

Internal problem ID [14414]
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(f)
Date solved : Thursday, October 02, 2025 at 09:36:53 AM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

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

✓ Maple. Time used: 0.011 (sec). Leaf size: 52

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

\[ x = {\mathrm e}^{-\frac {t}{2}} \sin \left (\frac {\sqrt {3}\, t}{2}\right ) c_2 +{\mathrm e}^{-\frac {t}{2}} \cos \left (\frac {\sqrt {3}\, t}{2}\right ) c_1 +t^{2}-2 t +\frac {{\mathrm e}^{2 t} \left (5 \sin \left (t \right )+6 \cos \left (t \right )\right )}{61} \]

✓ Mathematica. Time used: 0.93 (sec). Leaf size: 76

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

\begin{align*} x(t)&\to t^2-2 t+\frac {5}{61} e^{2 t} \sin (t)+\frac {6}{61} e^{2 t} \cos (t)+c_2 e^{-t/2} \cos \left (\frac {\sqrt {3} t}{2}\right )+c_1 e^{-t/2} \sin \left (\frac {\sqrt {3} t}{2}\right ) \end{align*}

✓ Sympy. Time used: 0.183 (sec). Leaf size: 54

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

\[ x{\left (t \right )} = t^{2} - 2 t + \left (C_{1} \sin {\left (\frac {\sqrt {3} t}{2} \right )} + C_{2} \cos {\left (\frac {\sqrt {3} t}{2} \right )}\right ) e^{- \frac {t}{2}} + \frac {\left (5 \sin {\left (t \right )} + 6 \cos {\left (t \right )}\right ) e^{2 t}}{61} \]