82.4.6 problem 28-6

Internal problem ID [21824]
Book : The Differential Equations Problem Solver. VOL. II. M. Fogiel director. REA, NY. 1978. ISBN 78-63609
Section : Chapter 28. Laplace transforms. Page 850
Problem number : 28-6
Date solved : Thursday, October 02, 2025 at 08:02:36 PM
CAS classification : [[_linear, `class A`]]

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

Using Laplace method With initial conditions

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

✓ Maple. Time used: 0.040 (sec). Leaf size: 19

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

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

✓ Mathematica. Time used: 0.026 (sec). Leaf size: 23

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

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

✓ Sympy. Time used: 0.080 (sec). Leaf size: 19

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

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