57.15.2 problem 6(b)

Internal problem ID [14466]
Book : A First Course in Differential Equations by J. David Logan. Third Edition. Springer-Verlag, NY. 2015.
Section : Chapter 3, Laplace transform. Section 3.2.1 Initial value problems. Exercises page 156
Problem number : 6(b)
Date solved : Thursday, October 02, 2025 at 09:37:33 AM
CAS classification : [[_linear, `class A`]]

\begin{align*} x^{\prime }+x&=\sin \left (2 t \right ) \end{align*}

Using Laplace method With initial conditions

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

✓ Maple. Time used: 0.110 (sec). Leaf size: 23

ode:=diff(x(t),t)+x(t) = sin(2*t); 
ic:=[x(0) = 0]; 
dsolve([ode,op(ic)],x(t),method='laplace');
 

\[ x = \frac {2 \,{\mathrm e}^{-t}}{5}-\frac {2 \cos \left (2 t \right )}{5}+\frac {\sin \left (2 t \right )}{5} \]

✓ Mathematica. Time used: 0.051 (sec). Leaf size: 27

ode=D[x[t],t]+x[t]==Sin[2*t]; 
ic={x[0]==0}; 
DSolve[{ode,ic},x[t],t,IncludeSingularSolutions->True]
 

\begin{align*} x(t)&\to \frac {1}{5} \left (2 e^{-t}+\sin (2 t)-2 \cos (2 t)\right ) \end{align*}

✓ Sympy. Time used: 0.091 (sec). Leaf size: 24

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

\[ x{\left (t \right )} = \frac {\sin {\left (2 t \right )}}{5} - \frac {2 \cos {\left (2 t \right )}}{5} + \frac {2 e^{- t}}{5} \]