84.38.7 problem 26.19

Internal problem ID [22361]
Book : Schaums outline series. Differential Equations By Richard Bronson. 1973. McGraw-Hill Inc. ISBN 0-07-008009-7
Section : Chapter 26. Solutions of linear differential equations with constant coefficients by Laplace transform. Supplementary problems
Problem number : 26.19
Date solved : Thursday, October 02, 2025 at 08:37:55 PM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

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

Using Laplace method With initial conditions

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

✓ Maple. Time used: 0.100 (sec). Leaf size: 13

ode:=diff(diff(y(t),t),t)-y(t) = sin(t); 
ic:=[y(0) = 0, D(y)(0) = 1]; 
dsolve([ode,op(ic)],y(t),method='laplace');
 

\[ y = -\frac {\sin \left (t \right )}{2}+\frac {3 \sinh \left (t \right )}{2} \]

✓ Mathematica. Time used: 0.033 (sec). Leaf size: 26

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

\begin{align*} y(t)&\to \frac {1}{4} \left (-3 e^{-t}+3 e^t-2 \sin (t)\right ) \end{align*}

✓ Sympy. Time used: 0.051 (sec). Leaf size: 20

from sympy import * 
t = symbols("t") 
y = Function("y") 
ode = Eq(-y(t) - sin(t) + Derivative(y(t), (t, 2)),0) 
ics = {y(0): 0, Subs(Derivative(y(t), t), t, 0): 1} 
dsolve(ode,func=y(t),ics=ics)
 

\[ y{\left (t \right )} = \frac {3 e^{t}}{4} - \frac {\sin {\left (t \right )}}{2} - \frac {3 e^{- t}}{4} \]