39.5.5 problem Problem 24.27

Internal problem ID [6547]
Book : Schaums Outline Differential Equations, 4th edition. Bronson and Costa. McGraw Hill 2014
Section : Chapter 24. Solutions of linear DE by Laplace transforms. Supplementary Problems. page 248
Problem number : Problem 24.27
Date solved : Sunday, March 30, 2025 at 11:07:04 AM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} y^{\prime \prime }-y&=\sin \left (x \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: 0.103 (sec). Leaf size: 13

ode:=diff(diff(y(x),x),x)-y(x) = sin(x); 
ic:=y(0) = 0, D(y)(0) = 1; 
dsolve([ode,ic],y(x),method='laplace');
 

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

✓ Mathematica. Time used: 0.032 (sec). Leaf size: 26

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

\[ y(x)\to \frac {1}{4} \left (-e^{-x}+5 e^x-2 \sin (x)\right ) \]

✓ Sympy. Time used: 0.121 (sec). Leaf size: 20

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

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