57.9.16 problem 2(d)

Internal problem ID [14424]
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 : 2(d)
Date solved : Thursday, October 02, 2025 at 09:37:01 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} x^{\prime \prime }-3 x^{\prime }-4 x&=2 t^{2} \end{align*}

✓ Maple. Time used: 0.003 (sec). Leaf size: 26

ode:=diff(diff(x(t),t),t)-3*diff(x(t),t)-4*x(t) = 2*t^2; 
dsolve(ode,x(t), singsol=all);
 

\[ x = {\mathrm e}^{-t} c_2 +{\mathrm e}^{4 t} c_1 -\frac {t^{2}}{2}+\frac {3 t}{4}-\frac {13}{16} \]

✓ Mathematica. Time used: 0.009 (sec). Leaf size: 37

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

\begin{align*} x(t)&\to -\frac {t^2}{2}+\frac {3 t}{4}+c_1 e^{-t}+c_2 e^{4 t}-\frac {13}{16} \end{align*}

✓ Sympy. Time used: 0.113 (sec). Leaf size: 27

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

\[ x{\left (t \right )} = C_{1} e^{- t} + C_{2} e^{4 t} - \frac {t^{2}}{2} + \frac {3 t}{4} - \frac {13}{16} \]