14.17.1 problem 19

Internal problem ID [2679]
Book : Differential equations and their applications, 4th ed., M. Braun
Section : Chapter 2. Second order differential equations. Section 2.10, Some useful properties of Laplace transform. Excercises page 238
Problem number : 19
Date solved : Sunday, March 30, 2025 at 12:13:55 AM
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 )&=1\\ y^{\prime }\left (0\right )&=2 \end{align*}

✓ Maple. Time used: 0.109 (sec). Leaf size: 16

ode:=diff(diff(y(t),t),t)+y(t) = sin(t); 
ic:=y(0) = 1, D(y)(0) = 2; 
dsolve([ode,ic],y(t),method='laplace');
 

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

✓ Mathematica. Time used: 0.033 (sec). Leaf size: 21

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

\[ y(t)\to \frac {5 \sin (t)}{2}-\frac {1}{2} t \cos (t)+\cos (t) \]

✓ Sympy. Time used: 0.098 (sec). Leaf size: 17

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

\[ y{\left (t \right )} = \left (1 - \frac {t}{2}\right ) \cos {\left (t \right )} + \frac {5 \sin {\left (t \right )}}{2} \]