71.15.5 problem 4 (e)

Internal problem ID [14476]
Book : Ordinary Differential Equations by Charles E. Roberts, Jr. CRC Press. 2010
Section : Chapter 5. The Laplace Transform Method. Exercises 5.4, page 265
Problem number : 4 (e)
Date solved : Monday, March 31, 2025 at 12:28:05 PM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} y^{\prime \prime }+4 y&=\left \{\begin {array}{cc} 0 & 0\le x <\pi \\ -\sin \left (3 x \right ) & \pi \le x \end {array}\right . \end{align*}

Using Laplace method With initial conditions

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

✓ Maple. Time used: 0.334 (sec). Leaf size: 35

ode:=diff(diff(y(x),x),x)+4*y(x) = piecewise(0 <= x and x < Pi,0,Pi <= x,-sin(3*x)); 
ic:=y(0) = 1, D(y)(0) = 1; 
dsolve([ode,ic],y(x),method='laplace');
 

\[ y = \cos \left (2 x \right )+\left (\left \{\begin {array}{cc} \frac {\sin \left (2 x \right )}{2} & x <\pi \\ \frac {4 \sin \left (2 x \right )}{5}+\frac {\sin \left (3 x \right )}{5} & \pi \le x \end {array}\right .\right ) \]

✓ Mathematica. Time used: 0.029 (sec). Leaf size: 42

ode=D[y[x],{x,2}]+4*y[x]==Piecewise[{ {0,0<=x<Pi},{Sin[3*(x-Pi)],x>=Pi}}]; 
ic={y[0]==1,Derivative[1][y][0] ==1}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\[ y(x)\to \begin {array}{cc} \{ & \begin {array}{cc} \cos (2 x)+\cos (x) \sin (x) & x\leq \pi \\ \frac {1}{5} (5 \cos (2 x)+4 \sin (2 x)+\sin (3 x)) & \text {True} \\ \end {array} \\ \end {array} \]

✓ Sympy. Time used: 0.500 (sec). Leaf size: 24

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(-Piecewise((0, (x >= 0) & (x < pi)), (-sin(3*x), x >= pi)) + 4*y(x) + Derivative(y(x), (x, 2)),0) 
ics = {y(0): 1, Subs(Derivative(y(x), x), x, 0): 1} 
dsolve(ode,func=y(x),ics=ics)
 

\[ y{\left (x \right )} = \begin {cases} 0 & \text {for}\: x \geq 0 \wedge x < \pi \\\frac {\sin {\left (3 x \right )}}{5} & \text {for}\: x \geq \pi \\\text {NaN} & \text {otherwise} \end {cases} + \frac {\sin {\left (2 x \right )}}{2} + \cos {\left (2 x \right )} \]