74.14.6 problem 6

Internal problem ID [16411]
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 : Tuesday, January 28, 2025 at 09:07:38 AM
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*}

✓ Solution by Maple

Time used: 0.005 (sec). Leaf size: 50

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

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

✓ Solution by Mathematica

Time used: 0.149 (sec). Leaf size: 138

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

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