68.14.6 problem 6

Internal problem ID [17698]
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 : 6
Date solved : Thursday, October 02, 2025 at 02:27:08 PM
CAS classification : [[_3rd_order, _linear, _nonhomogeneous]]

\begin{align*} y^{\prime \prime \prime }+10 y^{\prime \prime }+34 y^{\prime }+40 y&=t \,{\mathrm e}^{-4 t}+2 \,{\mathrm e}^{-3 t} \cos \left (t \right ) \end{align*}

✓ Maple. Time used: 0.009 (sec). Leaf size: 50

ode:=diff(diff(diff(y(t),t),t),t)+10*diff(diff(y(t),t),t)+34*diff(y(t),t)+40*y(t) = t*exp(-4*t)+2*exp(-3*t)*cos(t); 
dsolve(ode,y(t), singsol=all);
 

\[ y = \frac {\left (\left (-2 t +4 c_2 +4\right ) \cos \left (t \right )+2 \sin \left (t \right ) \left (t +2 c_3 +1\right )\right ) {\mathrm e}^{-3 t}}{4}+\frac {{\mathrm e}^{-4 t} \left (t^{2}+4 c_1 +2 t +1\right )}{4} \]

✓ Mathematica. Time used: 0.09 (sec). Leaf size: 138

ode=D[ y[t],{t,3}]+10*D[y[t],{t,2}]+34*D[y[t],t]+40*y[t]==t*Exp[-4*t]+2*Exp[-3*t]*Cos[t]; 
ic={}; 
DSolve[{ode,ic},y[t],t,IncludeSingularSolutions->True]
 

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

✓ Sympy. Time used: 0.422 (sec). Leaf size: 37

from sympy import * 
t = symbols("t") 
y = Function("y") 
ode = Eq(-t*exp(-4*t) + 40*y(t) + 34*Derivative(y(t), t) + 10*Derivative(y(t), (t, 2)) + Derivative(y(t), (t, 3)) - 2*exp(-3*t)*cos(t),0) 
ics = {} 
dsolve(ode,func=y(t),ics=ics)
 

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