50.25.4 problem 3(d)

Internal problem ID [8176]
Book : Differential Equations: Theory, Technique, and Practice by George Simmons, Steven Krantz. McGraw-Hill NY. 2007. 1st Edition.
Section : Chapter 7. Laplace Transforms. Section 7.5 Problesm for review and discovery. Section A, Drill exercises. Page 309
Problem number : 3(d)
Date solved : Monday, January 27, 2025 at 03:45:15 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} y^{\prime \prime }-y^{\prime }+y&=3 \,{\mathrm e}^{-t} \end{align*}

Using Laplace method With initial conditions

\begin{align*} y \left (0\right )&=3\\ y^{\prime }\left (0\right )&=2 \end{align*}

✓ Solution by Maple

Time used: 0.719 (sec). Leaf size: 41

dsolve([diff(y(t),t$2)-diff(y(t),t)+y(t)=3*exp(-t),y(0) = 3, D(y)(0) = 2],y(t), singsol=all)
 

\[ y = \frac {\left (4 \sin \left (\frac {\sqrt {3}\, t}{2}\right ) {\mathrm e}^{\frac {3 t}{2}} \sqrt {3}+6 \,{\mathrm e}^{\frac {3 t}{2}} \cos \left (\frac {\sqrt {3}\, t}{2}\right )+3\right ) {\mathrm e}^{-t}}{3} \]

✓ Solution by Mathematica

Time used: 0.028 (sec). Leaf size: 56

DSolve[{D[y[t],{t,2}]-D[y[t],t]+y[t]==3*Exp[-t],{y[0]==3,Derivative[1][y][0] ==2}},y[t],t,IncludeSingularSolutions -> True]
 

\[ y(t)\to e^{-t}+\frac {4 e^{t/2} \sin \left (\frac {\sqrt {3} t}{2}\right )}{\sqrt {3}}+2 e^{t/2} \cos \left (\frac {\sqrt {3} t}{2}\right ) \]