14.22.6 problem Problem 32

Internal problem ID [3961]
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
Date solved : Tuesday, September 30, 2025 at 06:59:29 AM
CAS classification : [[_linear, `class A`]]

\begin{align*} y^{\prime }-3 y&=\left \{\begin {array}{cc} \sin \left (t \right ) & 0\le t <\frac {\pi }{2} \\ 1 & \frac {\pi }{2}\le t \end {array}\right . \end{align*}

Using Laplace method With initial conditions

\begin{align*} y \left (0\right )&=2 \\ \end{align*}

✓ Maple. Time used: 0.358 (sec). Leaf size: 63

ode:=diff(y(t),t)-3*y(t) = piecewise(0 <= t and t < 1/2*Pi,sin(t),1/2*Pi <= t,1); 
ic:=[y(0) = 2]; 
dsolve([ode,op(ic)],y(t),method='laplace');
 

\[ y = \frac {\left (\left \{\begin {array}{cc} 21 \,{\mathrm e}^{3 t}-\cos \left (t \right )-3 \sin \left (t \right ) & t <\frac {\pi }{2} \\ -\frac {19}{3}+21 \,{\mathrm e}^{\frac {3 \pi }{2}} & t =\frac {\pi }{2} \\ 21 \,{\mathrm e}^{3 t}+\frac {{\mathrm e}^{3 t -\frac {3 \pi }{2}}}{3}-\frac {10}{3} & \frac {\pi }{2}<t \end {array}\right .\right )}{10} \]

✓ Mathematica. Time used: 0.063 (sec). Leaf size: 68

ode=D[y[t],t]-3*y[t]==Piecewise[{{Sin[t],0<=t<Pi/2},{1,t >= Pi/2}}]; 
ic={y[0]==2}; 
DSolve[{ode,ic},y[t],t,IncludeSingularSolutions->True]
 

\begin{align*} y(t)&\to \begin {array}{cc} \{ & \begin {array}{cc} 2 e^{3 t} & t\leq 0 \\ \frac {1}{30} \left (-10+63 e^{3 t}+e^{3 t-\frac {3 \pi }{2}}\right ) & 2 t>\pi \\ \frac {1}{10} \left (-\cos (t)+21 e^{3 t}-3 \sin (t)\right ) & \text {True} \\ \end {array} \\ \end {array} \end{align*}

✗ Sympy

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