20.22.12 problem Problem 38

Internal problem ID [3967]
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 38
Date solved : Sunday, March 30, 2025 at 02:13:09 AM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} y^{\prime \prime }+3 y^{\prime }+2 y&=-10 \operatorname {Heaviside}\left (t -\frac {\pi }{4}\right ) \cos \left (t +\frac {\pi }{4}\right ) \end{align*}

Using Laplace method With initial conditions

\begin{align*} y \left (0\right )&=1\\ y^{\prime }\left (0\right )&=0 \end{align*}

✓ Maple. Time used: 0.378 (sec). Leaf size: 67

ode:=diff(diff(y(t),t),t)+3*diff(y(t),t)+2*y(t) = -10*Heaviside(t-1/4*Pi)*cos(t+1/4*Pi); 
ic:=y(0) = 1, D(y)(0) = 0; 
dsolve([ode,ic],y(t),method='laplace');
 

\[ y = -2 \operatorname {Heaviside}\left (t -\frac {\pi }{4}\right ) {\mathrm e}^{\frac {\pi }{2}-2 t}+5 \operatorname {Heaviside}\left (t -\frac {\pi }{4}\right ) {\mathrm e}^{\frac {\pi }{4}-t}-2 \left (\cos \left (t \right )+\frac {\sin \left (t \right )}{2}\right ) \sqrt {2}\, \operatorname {Heaviside}\left (t -\frac {\pi }{4}\right )-{\mathrm e}^{-2 t}+2 \,{\mathrm e}^{-t} \]

✓ Mathematica. Time used: 0.131 (sec). Leaf size: 87

ode=D[y[t],{t,2}]+3*D[y[t],t]+2*y[t]==10*UnitStep[t-Pi/4]*Sin[t-Pi/4]; 
ic={y[0]==1,Derivative[1][y][0] ==0}; 
DSolve[{ode,ic},y[t],t,IncludeSingularSolutions->True]
 

\[ y(t)\to \begin {array}{cc} \{ & \begin {array}{cc} e^{-2 t} \left (-1+2 e^t\right ) & 4 t\leq \pi \\ -e^{-2 t} \left (2 \sqrt {2} e^{2 t} \cos (t)-2 e^t-5 e^{t+\frac {\pi }{4}}+\sqrt {2} e^{2 t} \sin (t)+2 e^{\pi /2}+1\right ) & \text {True} \\ \end {array} \\ \end {array} \]

✓ Sympy. Time used: 1.109 (sec). Leaf size: 82

from sympy import * 
t = symbols("t") 
y = Function("y") 
ode = Eq(2*y(t) + 10*cos(t + pi/4)*Heaviside(t - pi/4) + 3*Derivative(y(t), t) + Derivative(y(t), (t, 2)),0) 
ics = {y(0): 1, Subs(Derivative(y(t), t), t, 0): 0} 
dsolve(ode,func=y(t),ics=ics)
 

\[ y{\left (t \right )} = \left (5 e^{\frac {\pi }{4}} \theta \left (t - \frac {\pi }{4}\right ) + 2\right ) e^{- t} + \left (- 2 e^{\frac {\pi }{2}} \theta \left (t - \frac {\pi }{4}\right ) - 1\right ) e^{- 2 t} - \sqrt {2} \sin {\left (t \right )} \theta \left (t - \frac {\pi }{4}\right ) - 2 \sqrt {2} \cos {\left (t \right )} \theta \left (t - \frac {\pi }{4}\right ) \]