71.15.5 problem 4 (e)

Internal problem ID [14547]
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 : Tuesday, January 28, 2025 at 06:43:15 AM
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*}

✓ Solution by Maple

Time used: 12.750 (sec). Leaf size: 35

dsolve([diff(y(x),x$2)+4*y(x)=piecewise(0<=x and x<Pi,0,Pi<=x,sin(3*(x-Pi))),y(0) = 1, D(y)(0) = 1],y(x), singsol=all)
 

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

✓ Solution by Mathematica

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

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

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