Internal problem ID [12811]
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).
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _linear, _nonhomogeneous]]
\[ \boxed {y^{\prime \prime }+4 y=\left \{\begin {array}{cc} 0 & 0\le x <\pi \\ -\sin \left (3 x \right ) & \pi \le x \end {array}\right .} \] With initial conditions \begin {align*} [y \left (0\right ) = 1, y^{\prime }\left (0\right ) = 1] \end {align*}
✓ Solution by Maple
Time used: 8.188 (sec). Leaf size: 39
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)
\[ y \left (x \right ) = \cos \left (2 x \right )+\left (\left \{\begin {array}{cc} \frac {\sin \left (2 x \right )}{2} & x <\pi \\ \frac {4 \sin \left (2 x \right )}{5}+\frac {\sin \left (3 x \right )}{5} & \pi \le x \end {array}\right .\right ) \]
✓ Solution by Mathematica
Time used: 0.058 (sec). Leaf size: 42
DSolve[{y''[x]+4*y[x]==Piecewise[{ {0,0<=x<Pi},{Sin[3*(x-Pi)],x>=Pi}}],{y[0]==1,y'[0]==1}},y[x],x,IncludeSingularSolutions -> True]
\[ y(x)\to \begin {array}{cc} \{ & \begin {array}{cc} \cos (2 x)+\cos (x) \sin (x) & x\leq \pi \\ \frac {1}{5} (5 \cos (2 x)+4 \sin (2 x)+\sin (3 x)) & \text {True} \\ \end {array} \\ \end {array} \]