67.4.12 problem Problem 2(k)[l]

Internal problem ID [14056]
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(k)[l]
Date solved : Tuesday, January 28, 2025 at 06:13:18 AM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} 2 y^{\prime \prime }+y^{\prime }-y&=4 \sin \left (t \right ) \end{align*}

Using Laplace method With initial conditions

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

✓ Solution by Maple

Time used: 8.079 (sec). Leaf size: 28

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

\[ y = -\frac {2 \left (-5+\left (\cos \left (t \right )+3 \sin \left (t \right )\right ) {\mathrm e}^{t}+4 \,{\mathrm e}^{\frac {3 t}{2}}\right ) {\mathrm e}^{-t}}{5} \]

✓ Solution by Mathematica

Time used: 0.020 (sec). Leaf size: 34

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

\[ y(t)\to \frac {2}{5} \left (5 e^{-t}-4 e^{t/2}-3 \sin (t)-\cos (t)\right ) \]