68.14.7 problem 7

Internal problem ID [17699]
Book : INTRODUCTORY DIFFERENTIAL EQUATIONS. Martha L. Abell, James P. Braselton. Fourth edition 2014. ElScAe. 2014
Section : Chapter 4. Higher Order Equations. Exercises 4.6, page 187
Problem number : 7
Date solved : Thursday, October 02, 2025 at 02:27:09 PM
CAS classification : [[_3rd_order, _linear, _nonhomogeneous]]

\begin{align*} y^{\prime \prime \prime }+6 y^{\prime \prime }+11 y^{\prime }+6 y&=2 \,{\mathrm e}^{-3 t}-t \,{\mathrm e}^{-t} \end{align*}

✓ Maple. Time used: 0.005 (sec). Leaf size: 39

ode:=diff(diff(diff(y(t),t),t),t)+6*diff(diff(y(t),t),t)+11*diff(y(t),t)+6*y(t) = 2*exp(-3*t)-t*exp(-t); 
dsolve(ode,y(t), singsol=all);
 

\[ y = -\frac {\left (\left (t^{2}-3 t -4 c_3 +\frac {7}{2}\right ) {\mathrm e}^{2 t}-4 c_2 \,{\mathrm e}^{t}-4 t -4 c_1 -6\right ) {\mathrm e}^{-3 t}}{4} \]

✓ Mathematica. Time used: 0.06 (sec). Leaf size: 97

ode=D[ y[t],{t,3}]+6*D[y[t],{t,2}]+11*D[y[t],t]+6*y[t]==2*Exp[-3*t]-t*Exp[-t]; 
ic={}; 
DSolve[{ode,ic},y[t],t,IncludeSingularSolutions->True]
 

\begin{align*} y(t)&\to \frac {1}{4} e^{-3 t} \left (4 \int _1^t\left (1-\frac {1}{2} e^{2 K[1]} K[1]\right )dK[1]+4 e^t \int _1^t\left (e^{K[2]} K[2]-2 e^{-K[2]}\right )dK[2]-e^{2 t} t^2+4 c_2 e^t+4 c_3 e^{2 t}-2+4 c_1\right ) \end{align*}

✓ Sympy. Time used: 0.270 (sec). Leaf size: 31

from sympy import * 
t = symbols("t") 
y = Function("y") 
ode = Eq(t*exp(-t) + 6*y(t) + 11*Derivative(y(t), t) + 6*Derivative(y(t), (t, 2)) + Derivative(y(t), (t, 3)) - 2*exp(-3*t),0) 
ics = {} 
dsolve(ode,func=y(t),ics=ics)
 

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