67.4.5 problem Problem 2(e)

Internal problem ID [14049]
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(e)
Date solved : Tuesday, January 28, 2025 at 06:13:13 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} y^{\prime \prime }+9 y&={\mathrm e}^{-2 t} \end{align*}

Using Laplace method With initial conditions

\begin{align*} y \left (0\right )&=-{\frac {2}{13}}\\ y^{\prime }\left (0\right )&={\frac {1}{13}} \end{align*}

✓ Solution by Maple

Time used: 21.529 (sec). Leaf size: 23

dsolve([diff(y(t),t$2)+9*y(t)=exp(-2*t),y(0) = -2/13, D(y)(0) = 1/13],y(t), singsol=all)
 

\[ y = \frac {{\mathrm e}^{-2 t}}{13}-\frac {3 \cos \left (3 t \right )}{13}+\frac {\sin \left (3 t \right )}{13} \]

✓ Solution by Mathematica

Time used: 0.156 (sec). Leaf size: 132

DSolve[{D[y[t],{t,2}]+9*y[t]==Exp[-2*t],{y[0]==-2/13,Derivative[1][y][0] ==1/13}},y[t],t,IncludeSingularSolutions -> True]
 

\[ y(t)\to \cos (3 t) \left (-\int _1^0-\frac {1}{3} e^{-2 K[1]} \sin (3 K[1])dK[1]\right )+\cos (3 t) \int _1^t-\frac {1}{3} e^{-2 K[1]} \sin (3 K[1])dK[1]-\sin (3 t) \int _1^0\frac {1}{3} e^{-2 K[2]} \cos (3 K[2])dK[2]+\sin (3 t) \int _1^t\frac {1}{3} e^{-2 K[2]} \cos (3 K[2])dK[2]+\frac {1}{39} \sin (3 t)-\frac {2}{13} \cos (3 t) \]