20.22.7 problem Problem 33

Internal problem ID [3962]
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 33
Date solved : Monday, January 27, 2025 at 08:05:05 AM
CAS classification : [[_linear, `class A`]]

\begin{align*} y^{\prime }-3 y&=-10 \,{\mathrm e}^{-t +a} \sin \left (-2 t +2 a \right ) \operatorname {Heaviside}\left (t -a \right ) \end{align*}

Using Laplace method With initial conditions

\begin{align*} y \left (0\right )&=5 \end{align*}

✓ Solution by Maple

Time used: 6.211 (sec). Leaf size: 101

dsolve([diff(y(t),t)-3*y(t)=10*exp(-(t-a))*sin(2*(t-a))*Heaviside(t-a),y(0) = 5],y(t), singsol=all)
 

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

✓ Solution by Mathematica

Time used: 0.469 (sec). Leaf size: 103

DSolve[{D[y[t],t]-3*y[t]==10*Exp[-(t-a)]*Sin[2*(t-a)]*UnitStep[t-a],{y[0]==5}},y[t],t,IncludeSingularSolutions -> True]
 

\[ y(t)\to e^{-3 a-t} \left (e^{4 t} \theta (-a) \left (-2 e^{4 a} \sin (2 a)+e^{4 a} \cos (2 a)-1\right )+\theta (t-a) \left (2 e^{4 a} \sin (2 (a-t))-e^{4 a} \cos (2 (a-t))+e^{4 t}\right )+5 e^{3 a+4 t}\right ) \]