73.19.8 problem 28.9 (a)

Internal problem ID [15606]
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 (a)
Date solved : Tuesday, January 28, 2025 at 08:03:05 AM
CAS classification : [[_2nd_order, _quadrature]]

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

Using Laplace method With initial conditions

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

✓ Solution by Maple

Time used: 8.608 (sec). Leaf size: 15

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

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

✓ Solution by Mathematica

Time used: 0.026 (sec). Leaf size: 72

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

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