72.21.1 problem 1

Internal problem ID [14971]
Book : DIFFERENTIAL EQUATIONS by Paul Blanchard, Robert L. Devaney, Glen R. Hall. 4th edition. Brooks/Cole. Boston, USA. 2012
Section : Chapter 6. Laplace transform. Section 6.6. page 624
Problem number : 1
Date solved : Tuesday, January 28, 2025 at 07:25:54 AM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

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

Using Laplace method With initial conditions

\begin{align*} y \left (0\right )&=2\\ y^{\prime }\left (0\right )&=-2 \end{align*}

✓ Solution by Maple

Time used: 10.433 (sec). Leaf size: 37

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

\[ y = \frac {{\mathrm e}^{-2 t} \left (-7 \sin \left (4 t \right )+4 \cos \left (4 t \right )\right )}{130}+\frac {128 \left (\cos \left (t \right )+\frac {\sin \left (t \right )}{8}\right ) {\mathrm e}^{-t}}{65} \]

✓ Solution by Mathematica

Time used: 0.131 (sec). Leaf size: 116

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

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