5.4.5 problem 5

Internal problem ID [1499]
Book : Elementary differential equations and boundary value problems, 11th ed., Boyce, DiPrima, Meade
Section : Chapter 6.4, The Laplace Transform. Differential equations with discontinuous forcing functions. page 268
Problem number : 5
Date solved : Tuesday, September 30, 2025 at 04:34:44 AM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} y^{\prime \prime }+y^{\prime }+\frac {5 y}{4}&=t -\operatorname {Heaviside}\left (t -\frac {\pi }{2}\right ) \left (t -\frac {\pi }{2}\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: 0.360 (sec). Leaf size: 63

ode:=diff(diff(y(t),t),t)+diff(y(t),t)+5/4*y(t) = t-Heaviside(t-1/2*Pi)*(t-1/2*Pi); 
ic:=[y(0) = 0, D(y)(0) = 0]; 
dsolve([ode,op(ic)],y(t),method='laplace');
 

\[ y = -\frac {16}{25}-\frac {12 \operatorname {Heaviside}\left (t -\frac {\pi }{2}\right ) \left (\cos \left (t \right )+\frac {4 \sin \left (t \right )}{3}\right ) {\mathrm e}^{-\frac {t}{2}+\frac {\pi }{4}}}{25}+\frac {2 \left (8-10 t +5 \pi \right ) \operatorname {Heaviside}\left (t -\frac {\pi }{2}\right )}{25}+\frac {4 \,{\mathrm e}^{-\frac {t}{2}} \left (4 \cos \left (t \right )-3 \sin \left (t \right )\right )}{25}+\frac {4 t}{5} \]

✓ Mathematica. Time used: 0.023 (sec). Leaf size: 96

ode=D[y[t],{t,2}]+D[y[t],t]+5/4*y[t]==t-UnitStep[t-Pi/2]*(t-Pi/2); 
ic={y[0]==0,Derivative[1][y][0] ==0}; 
DSolve[{ode,ic},y[t],t,IncludeSingularSolutions->True]
 

\begin{align*} y(t)&\to \begin {array}{cc} \{ & \begin {array}{cc} \frac {4}{25} e^{-t/2} \left (e^{t/2} (5 t-4)+4 \cos (t)-3 \sin (t)\right ) & 2 t\leq \pi \\ -\frac {2}{25} e^{-t/2} \left (\left (-8+6 e^{\pi /4}\right ) \cos (t)+\left (6+8 e^{\pi /4}\right ) \sin (t)-5 e^{t/2} \pi \right ) & \text {True} \\ \end {array} \\ \end {array} \end{align*}

✓ Sympy. Time used: 2.628 (sec). Leaf size: 104

from sympy import * 
t = symbols("t") 
y = Function("y") 
ode = Eq(-t + (t - pi/2)*Heaviside(t - pi/2) + 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)
 

\[ y{\left (t \right )} = - \frac {4 t \theta \left (t - \frac {\pi }{2}\right )}{5} + \frac {4 t}{5} + \left (\left (- \frac {16 e^{\frac {\pi }{4}} \theta \left (t - \frac {\pi }{2}\right )}{25} - \frac {12}{25}\right ) \sin {\left (t \right )} + \left (- \frac {12 e^{\frac {\pi }{4}} \theta \left (t - \frac {\pi }{2}\right )}{25} + \frac {16}{25}\right ) \cos {\left (t \right )}\right ) e^{- \frac {t}{2}} + \frac {16 \theta \left (t - \frac {\pi }{2}\right )}{25} + \frac {2 \pi \theta \left (t - \frac {\pi }{2}\right )}{5} - \frac {16}{25} \]