70.20.10 problem 10

Internal problem ID [19044]
Book : Differential equations. An introduction to modern methods and applications. James Brannan, William E. Boyce. Third edition. Wiley 2015
Section : Chapter 5. The Laplace transform. Section 5.6 (Differential equations with Discontinuous Forcing Functions). Problems at page 342
Problem number : 10
Date solved : Thursday, October 02, 2025 at 03:37:15 PM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} y^{\prime \prime }+y^{\prime }+\frac {5 y}{4}&=\left \{\begin {array}{cc} \sin \left (t \right ) & 0\le t <\pi \\ 0 & \pi \le t \end {array}\right . \end{align*}

Using Laplace method With initial conditions

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

✓ Maple. Time used: 0.297 (sec). Leaf size: 55

ode:=diff(diff(y(t),t),t)+diff(y(t),t)+5/4*y(t) = piecewise(0 <= t and t < Pi,sin(t),Pi <= t,0); 
ic:=[y(0) = 0, D(y)(0) = 0]; 
dsolve([ode,op(ic)],y(t),method='laplace');
 

\[ y = -\frac {8 \left (\left \{\begin {array}{cc} {\mathrm e}^{-\frac {t}{4}} \left (-\sin \left (t \right ) \cosh \left (\frac {t}{4}\right )+4 \cos \left (t \right ) \sinh \left (\frac {t}{4}\right )\right ) & t <\pi \\ \left (4 \cos \left (t \right )+\sin \left (t \right )\right ) \sinh \left (\frac {\pi }{4}\right ) {\mathrm e}^{-\frac {t}{2}+\frac {\pi }{4}} & \pi \le t \end {array}\right .\right )}{17} \]

✓ Mathematica. Time used: 117.297 (sec). Leaf size: 161

ode=D[y[t],{t,2}]+D[y[t],t]+125/100*y[t]==Piecewise[{  {Sin[t],0<= t <Pi}, {0,t>=Pi}}]; 
ic={y[0]==0,Derivative[1][y][0] ==0}; 
DSolve[{ode,ic},y[t],t,IncludeSingularSolutions->True]
 

\begin{align*} y(t)&\to e^{-t/2} \left (-\sin (t) \int _1^0e^{\frac {K[3]}{2}} \cos (K[3]) \left ( \begin {array}{cc} \{ & \begin {array}{cc} \sin (K[3]) & 0\leq K[3]<\pi \\ 0 & \text {True} \\ \end {array} \\ \end {array} \right )dK[3]+\sin (t) \int _1^te^{\frac {K[3]}{2}} \cos (K[3]) \left ( \begin {array}{cc} \{ & \begin {array}{cc} \sin (K[3]) & 0\leq K[3]<\pi \\ 0 & \text {True} \\ \end {array} \\ \end {array} \right )dK[3]+\cos (t) \left (\int _1^t-e^{\frac {K[4]}{2}} \left ( \begin {array}{cc} \{ & \begin {array}{cc} \sin (K[4]) & 0\leq K[4]<\pi \\ 0 & \text {True} \\ \end {array} \\ \end {array} \right ) \sin (K[4])dK[4]-\int _1^0-e^{\frac {K[4]}{2}} \left ( \begin {array}{cc} \{ & \begin {array}{cc} \sin (K[4]) & 0\leq K[4]<\pi \\ 0 & \text {True} \\ \end {array} \\ \end {array} \right ) \sin (K[4])dK[4]\right )\right ) \end{align*}

✗ Sympy

from sympy import * 
t = symbols("t") 
y = Function("y") 
ode = Eq(-Piecewise((sin(t), (t >= 0) & (t < pi)), (0, t >= pi)) + 5*y(t)/4 + Derivative(y(t), t) + Derivative(y(t), (t, 2)),0) 
ics = {y(0): 0, Subs(Derivative(y(t), t), t, 0): 0} 
dsolve(ode,func=y(t),ics=ics)