Internal problem ID [2363]
Book: Differential equations and linear algebra, Stephen W. Goode and Scott A Annin. Fourth edition,
2015
Section: Chapter 10, The Laplace Transform and Some Elementary Applications. Exercises for 10.7.
page 704
Problem number: Problem 32.
ODE order: 1.
ODE degree: 1.
CAS Maple gives this as type [[_linear, `class A`]]
\[ \boxed {y^{\prime }-3 y-\left (\left \{\begin {array}{cc} \sin \left (t \right ) & 0\le t <\frac {\pi }{2} \\ 1 & \frac {\pi }{2}\le t \end {array}\right .\right )=0} \] With initial conditions \begin {align*} [y \left (0\right ) = 2] \end {align*}
✓ Solution by Maple
Time used: 0.235 (sec). Leaf size: 57
dsolve([diff(y(t),t)-3*y(t)=piecewise(0<=t and t<Pi/2,sin(t),t>=Pi/2,1),y(0) = 2],y(t), singsol=all)
\[ y \left (t \right ) = \left \{\begin {array}{cc} 2 \,{\mathrm e}^{3 t} & t <0 \\ \frac {21 \,{\mathrm e}^{3 t}}{10}-\frac {\cos \left (t \right )}{10}-\frac {3 \sin \left (t \right )}{10} & t <\frac {\pi }{2} \\ \frac {21 \,{\mathrm e}^{3 t}}{10}+\frac {{\mathrm e}^{3 t -\frac {3 \pi }{2}}}{30}-\frac {1}{3} & \frac {\pi }{2}\le t \end {array}\right . \]
✓ Solution by Mathematica
Time used: 0.104 (sec). Leaf size: 65
DSolve[{y'[t]-3*y[t]==Piecewise[{{Sin[t],0<=t<Pi/2},{1,t >= Pi/2}}],{y[0]==2}},y[t],t,IncludeSingularSolutions -> True]
\begin{align*} y(t)\to {cc} \{ & {cc} 2 e^{3 t} & t\leq 0 \\ \frac {1}{30} \left (-10+e^{3 t} \left (63+e^{-3 \pi /2}\right )\right ) & 2 t>\pi \\ \frac {1}{10} \left (-\cos (t)+21 e^{3 t}-3 \sin (t)\right ) & \text {True} \\ \\ \\ \\ \\ \end{align*}