67.4.10 problem Problem 2(i)[j]

Internal problem ID [14054]
Book : APPLIED DIFFERENTIAL EQUATIONS The Primary Course by Vladimir A. Dobrushkin. CRC Press 2015
Section : Chapter 5.6 Laplace transform. Nonhomogeneous equations. Problems page 368
Problem number : Problem 2(i)[j]
Date solved : Tuesday, January 28, 2025 at 06:13:17 AM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} y^{\prime \prime }+8 y^{\prime }+20 y&=\sin \left (2 t \right ) \end{align*}

Using Laplace method With initial conditions

\begin{align*} y \left (0\right )&=1\\ y^{\prime }\left (0\right )&=-4 \end{align*}

✓ Solution by Maple

Time used: 8.566 (sec). Leaf size: 31

dsolve([diff(y(t),t$2)+8*diff(y(t),t)+20*y(t)=sin(2*t),y(0) = 1, D(y)(0) = -4],y(t), singsol=all)
 

\[ y = \frac {\left (-1+33 \,{\mathrm e}^{-4 t}\right ) \cos \left (2 t \right )}{32}+\frac {\sin \left (2 t \right ) \left (1+{\mathrm e}^{-4 t}\right )}{32} \]

✓ Solution by Mathematica

Time used: 0.275 (sec). Leaf size: 124

DSolve[{D[y[t],{t,2}]+8*D[y[t],t]+20*y[t]==Sin[2*t],{y[0]==1,Derivative[1][y][0] ==-4}},y[t],t,IncludeSingularSolutions -> True]
 

\[ y(t)\to e^{-4 t} \left (-\sin (2 t) \int _1^0\frac {1}{4} e^{4 K[1]} \sin (4 K[1])dK[1]+\sin (2 t) \int _1^t\frac {1}{4} e^{4 K[1]} \sin (4 K[1])dK[1]+\cos (2 t) \left (\int _1^t-\frac {1}{2} e^{4 K[2]} \sin ^2(2 K[2])dK[2]-\int _1^0-\frac {1}{2} e^{4 K[2]} \sin ^2(2 K[2])dK[2]+1\right )\right ) \]