73.19.11 problem 28.9 (d)

Internal problem ID [15609]
Book : Ordinary Differential Equations. An introduction to the fundamentals. Kenneth B. Howell. second edition. CRC Press. FL, USA. 2020
Section : Chapter 28. The inverse Laplace transform. Additional Exercises. page 509
Problem number : 28.9 (d)
Date solved : Tuesday, January 28, 2025 at 08:03:07 AM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} y^{\prime \prime }-4 y^{\prime }+13 y&={\mathrm e}^{2 t} \sin \left (3 t \right ) \end{align*}

Using Laplace method With initial conditions

\begin{align*} y \left (0\right )&=4\\ y^{\prime }\left (0\right )&=3 \end{align*}

✓ Solution by Maple

Time used: 7.659 (sec). Leaf size: 28

dsolve([diff(y(t),t$2)-4*diff(y(t),t)+13*y(t)=exp(2*t)*sin(3*t),y(0) = 4, D(y)(0) = 3],y(t), singsol=all)
 

\[ y = -\frac {\left (-24+t \right ) {\mathrm e}^{2 t} \cos \left (3 t \right )}{6}-\frac {29 \,{\mathrm e}^{2 t} \sin \left (3 t \right )}{18} \]

✓ Solution by Mathematica

Time used: 0.057 (sec). Leaf size: 119

DSolve[{D[y[t],{t,2}]-4*D[y[t],t]+13*y[t]==Exp[2*t]*Sin[3*t],{y[0]==4,Derivative[1][y][0] ==3}},y[t],t,IncludeSingularSolutions -> True]
 

\[ y(t)\to \frac {1}{3} e^{2 t} \left (-3 \sin (3 t) \int _1^0\frac {1}{6} \sin (6 K[1])dK[1]+3 \sin (3 t) \int _1^t\frac {1}{6} \sin (6 K[1])dK[1]-3 \cos (3 t) \int _1^0-\frac {1}{3} \sin ^2(3 K[2])dK[2]+3 \cos (3 t) \int _1^t-\frac {1}{3} \sin ^2(3 K[2])dK[2]-5 \sin (3 t)+12 \cos (3 t)\right ) \]