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 : Monday, January 27, 2025 at 08:05:21 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*}

✓ Solution by Maple

Time used: 3.166 (sec). Leaf size: 67

dsolve([diff(y(t),t$2)+3*diff(y(t),t)+2*y(t)=10*Heaviside(t-Pi/4)*sin(t-Pi/4),y(0) = 1, D(y)(0) = 0],y(t), singsol=all)
 

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

✓ Solution by Mathematica

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

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

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