74.18.30 problem 36

Internal problem ID [16587]
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 : Tuesday, January 28, 2025 at 09:11:48 AM
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*}

✓ Solution by Maple

Time used: 0.009 (sec). Leaf size: 70

dsolve(diff(y(t),t$4)+6*diff(y(t),t$3)+18*diff(y(t),t$2)+30*diff(y(t),t)+25*y(t)=exp(-t)*cos(2*t)+exp(-2*t)*sin(t),y(t), singsol=all)
 

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

✓ Solution by Mathematica

Time used: 0.409 (sec). Leaf size: 222

DSolve[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],y[t],t,IncludeSingularSolutions -> True]
 

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