20.22.15 problem Problem 41

Internal problem ID [3970]
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 41
Date solved : Sunday, March 30, 2025 at 02:13:17 AM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} y^{\prime \prime }-2 y^{\prime }+5 y&=2 \sin \left (t \right )+\operatorname {Heaviside}\left (t -\frac {\pi }{2}\right ) \left (1+\cos \left (t \right )\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: 0.440 (sec). Leaf size: 72

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

\[ y = \frac {\left (\left (-1+2 \cos \left (t \right )^{2}-3 \sin \left (t \right ) \cos \left (t \right )\right ) {\mathrm e}^{t -\frac {\pi }{2}}+2 \cos \left (t \right )-\sin \left (t \right )+2\right ) \operatorname {Heaviside}\left (t -\frac {\pi }{2}\right )}{10}-\frac {2 \cos \left (t \right )^{2} {\mathrm e}^{t}}{5}-\frac {\cos \left (t \right ) {\mathrm e}^{t} \sin \left (t \right )}{5}+\frac {\cos \left (t \right )}{5}+\frac {{\mathrm e}^{t}}{5}+\frac {2 \sin \left (t \right )}{5} \]

✓ Mathematica. Time used: 0.441 (sec). Leaf size: 98

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

\[ y(t)\to \begin {array}{cc} \{ & \begin {array}{cc} \frac {1}{5} \left (-e^t \sin (t) \cos (t)+\cos (t)-e^t \cos (2 t)+2 \sin (t)\right ) & 2 t\leq \pi \\ \frac {1}{20} \left (8 \cos (t)+2 e^t \left (-2+e^{-\pi /2}\right ) \cos (2 t)+6 \sin (t)-2 e^t \sin (2 t)-3 e^{t-\frac {\pi }{2}} \sin (2 t)+4\right ) & \text {True} \\ \end {array} \\ \end {array} \]

✗ Sympy

from sympy import * 
t = symbols("t") 
y = Function("y") 
ode = Eq((-cos(t) - 1)*Heaviside(t - pi/2) + 5*y(t) - 2*sin(t) - 2*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)
 

Timed Out