68.14.22 problem 22

Internal problem ID [17714]
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 : 22
Date solved : Thursday, October 02, 2025 at 02:27:17 PM
CAS classification : [[_3rd_order, _linear, _nonhomogeneous]]

\begin{align*} y^{\prime \prime \prime }-13 y^{\prime }+12 y&=\cos \left (t \right ) \end{align*}

✓ Maple. Time used: 0.004 (sec). Leaf size: 29

ode:=diff(diff(diff(y(t),t),t),t)-13*diff(y(t),t)+12*y(t) = cos(t); 
dsolve(ode,y(t), singsol=all);
 

\[ y = \frac {3 \cos \left (t \right )}{85}-\frac {7 \sin \left (t \right )}{170}+c_1 \,{\mathrm e}^{t}+c_2 \,{\mathrm e}^{-4 t}+c_3 \,{\mathrm e}^{3 t} \]

✓ Mathematica. Time used: 0.018 (sec). Leaf size: 102

ode=D[ y[t],{t,3}]-13*D[y[t],t]+12*y[t]==Cos[t]; 
ic={}; 
DSolve[{ode,ic},y[t],t,IncludeSingularSolutions->True]
 

\begin{align*} y(t)&\to e^{-4 t} \left (\int _1^t\frac {1}{35} e^{4 K[1]} \cos (K[1])dK[1]+e^{5 t} \int _1^t-\frac {1}{10} e^{-K[2]} \cos (K[2])dK[2]+e^{7 t} \int _1^t\frac {1}{14} e^{-3 K[3]} \cos (K[3])dK[3]+c_2 e^{5 t}+c_3 e^{7 t}+c_1\right ) \end{align*}

✓ Sympy. Time used: 0.133 (sec). Leaf size: 34

from sympy import * 
t = symbols("t") 
y = Function("y") 
ode = Eq(12*y(t) - cos(t) - 13*Derivative(y(t), t) + Derivative(y(t), (t, 3)),0) 
ics = {} 
dsolve(ode,func=y(t),ics=ics)
 

\[ y{\left (t \right )} = C_{1} e^{- 4 t} + C_{2} e^{t} + C_{3} e^{3 t} - \frac {7 \sin {\left (t \right )}}{170} + \frac {3 \cos {\left (t \right )}}{85} \]