68.18.30 problem 36

Internal problem ID [17874]
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 : 36
Date solved : Thursday, October 02, 2025 at 02:29:05 PM
CAS classification : [[_high_order, _linear, _nonhomogeneous]]

\begin{align*} y^{\prime \prime \prime \prime }+6 y^{\prime \prime \prime }+18 y^{\prime \prime }+30 y^{\prime }+25 y&={\mathrm e}^{-t} \cos \left (2 t \right )+{\mathrm e}^{-2 t} \sin \left (t \right ) \end{align*}

✓ Maple. Time used: 0.011 (sec). Leaf size: 70

ode:=diff(diff(diff(diff(y(t),t),t),t),t)+6*diff(diff(diff(y(t),t),t),t)+18*diff(diff(y(t),t),t)+30*diff(y(t),t)+25*y(t) = exp(-t)*cos(2*t)+exp(-2*t)*sin(t); 
dsolve(ode,y(t), singsol=all);
 

\[ y = \frac {\left (\left (-20 t +400 c_3 -6\right ) \cos \left (t \right )^{2}-10 \sin \left (t \right ) \left (t -40 c_4 -\frac {21}{5}\right ) \cos \left (t \right )+10 t -200 c_3 +3\right ) {\mathrm e}^{-t}}{200}-\frac {\left (\left (t -10 c_1 +\frac {7}{10}\right ) \cos \left (t \right )-\frac {\left (t +20 c_2 +\frac {1}{5}\right ) \sin \left (t \right )}{2}\right ) {\mathrm e}^{-2 t}}{10} \]

✓ Mathematica. Time used: 0.237 (sec). Leaf size: 222

ode=D[y[t],{t,4}]+6*D[ y[t],{t,3}]+18*D[y[t],{t,2}]+30*D[y[t],t]+25*y[t]==Exp[-t]*Cos[2*t]+Exp[-2*t]*Sin[t]; 
ic={}; 
DSolve[{ode,ic},y[t],t,IncludeSingularSolutions->True]
 

\begin{align*} y(t)&\to e^{-2 t} \left (\cos (t) \int _1^t\frac {1}{10} (\cos (K[2])-2 \sin (K[2])) \left (e^{K[2]} \cos (2 K[2])+\sin (K[2])\right )dK[2]+\sin (t) \int _1^t\frac {1}{10} (2 \cos (K[1])+\sin (K[1])) \left (e^{K[1]} \cos (2 K[1])+\sin (K[1])\right )dK[1]+e^t \sin (2 t) \int _1^t-\frac {1}{20} e^{-K[3]} \left (e^{K[3]} \cos (2 K[3])+\sin (K[3])\right ) (\cos (2 K[3])+2 \sin (2 K[3]))dK[3]+e^t \cos (2 t) \int _1^t-\frac {1}{20} e^{-K[4]} \left (e^{K[4]} \cos (2 K[4])+\sin (K[4])\right ) (2 \cos (2 K[4])-\sin (2 K[4]))dK[4]+c_2 \cos (t)+c_4 e^t \cos (2 t)+c_1 \sin (t)+c_3 e^t \sin (2 t)\right ) \end{align*}

✓ Sympy. Time used: 0.636 (sec). Leaf size: 46

from sympy import * 
t = symbols("t") 
y = Function("y") 
ode = Eq(25*y(t) + 30*Derivative(y(t), t) + 18*Derivative(y(t), (t, 2)) + 6*Derivative(y(t), (t, 3)) + Derivative(y(t), (t, 4)) - exp(-t)*cos(2*t) - exp(-2*t)*sin(t),0) 
ics = {} 
dsolve(ode,func=y(t),ics=ics)
 

\[ y{\left (t \right )} = \left (\left (C_{1} - \frac {t}{20}\right ) \cos {\left (2 t \right )} + \left (C_{2} - \frac {t}{40}\right ) \sin {\left (2 t \right )} + \left (\left (C_{3} - \frac {t}{10}\right ) \cos {\left (t \right )} + \left (C_{4} + \frac {t}{20}\right ) \sin {\left (t \right )}\right ) e^{- t}\right ) e^{- t} \]