68.18.29 problem 35

Internal problem ID [17873]
Book : INTRODUCTORY DIFFERENTIAL EQUATIONS. Martha L. Abell, James P. Braselton. Fourth edition 2014. ElScAe. 2014
Section : Chapter 4. Higher Order Equations. Chapter 4 review exercises, page 219
Problem number : 35
Date solved : Thursday, October 02, 2025 at 02:29:05 PM
CAS classification : [[_3rd_order, _linear, _nonhomogeneous]]

\begin{align*} y^{\prime \prime \prime }-12 y^{\prime }-16 y&={\mathrm e}^{4 t}-{\mathrm e}^{-2 t} \end{align*}

✓ Maple. Time used: 0.004 (sec). Leaf size: 41

ode:=diff(diff(diff(y(t),t),t),t)-12*diff(y(t),t)-16*y(t) = exp(4*t)-exp(-2*t); 
dsolve(ode,y(t), singsol=all);
 

\[ y = \frac {{\mathrm e}^{-2 t} \left (\left (-1+108 c_2 +3 t \right ) {\mathrm e}^{6 t}+\frac {1}{2}+9 t^{2}+3 \left (1+36 c_3 \right ) t +108 c_1 \right )}{108} \]

✓ Mathematica. Time used: 0.058 (sec). Leaf size: 66

ode=D[ y[t],{t,3}]-12*D[y[t],t]-16*y[t]==Exp[4*t]-Exp[-2*t]; 
ic={}; 
DSolve[{ode,ic},y[t],t,IncludeSingularSolutions->True]
 

\begin{align*} y(t)&\to \frac {1}{216} e^{-2 t} \left (216 \int _1^t\frac {1}{36} \left (-1+e^{6 K[1]}\right ) (6 K[1]-1)dK[1]+36 t^2+216 c_2 t+216 c_3 e^{6 t}+1+216 c_1\right ) \end{align*}

✓ Sympy. Time used: 0.263 (sec). Leaf size: 26

from sympy import * 
t = symbols("t") 
y = Function("y") 
ode = Eq(-16*y(t) - exp(4*t) - 12*Derivative(y(t), t) + Derivative(y(t), (t, 3)) + exp(-2*t),0) 
ics = {} 
dsolve(ode,func=y(t),ics=ics)
 

\[ y{\left (t \right )} = \left (C_{1} + \frac {t}{36}\right ) e^{4 t} + \left (C_{2} + t \left (C_{3} + \frac {t}{12}\right )\right ) e^{- 2 t} \]