67.19.8 problem 28.9 (a)

Internal problem ID [16893]
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 : Thursday, October 02, 2025 at 01:40:10 PM
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*}

✓ Maple. Time used: 0.124 (sec). Leaf size: 15

ode:=diff(diff(y(t),t),t) = exp(t)*sin(t); 
ic:=[y(0) = 0, D(y)(0) = 0]; 
dsolve([ode,op(ic)],y(t),method='laplace');
 

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

✓ Mathematica. Time used: 0.016 (sec). Leaf size: 72

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

\begin{align*} 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] \end{align*}

✓ Sympy. Time used: 0.054 (sec). Leaf size: 17

from sympy import * 
t = symbols("t") 
y = Function("y") 
ode = Eq(-exp(t)*sin(t) + Derivative(y(t), (t, 2)),0) 
ics = {y(0): 0, Subs(Derivative(y(t), t), t, 0): 0} 
dsolve(ode,func=y(t),ics=ics)
 

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