50.25.2 problem 3(b)

Internal problem ID [8174]
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(b)
Date solved : Monday, January 27, 2025 at 03:45:13 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} y^{\prime \prime }+3 y^{\prime }-2 y&=-6 \,{\mathrm e}^{\pi -t} \end{align*}

Using Laplace method With initial conditions

\begin{align*} y \left (\pi \right )&=1\\ y^{\prime }\left (\pi \right )&=4 \end{align*}

✓ Solution by Maple

Time used: 0.717 (sec). Leaf size: 58

dsolve([diff(y(t),t$2)+3*diff(y(t),t)-2*y(t)=-6*exp(Pi-t),y(Pi) = 1, D(y)(Pi) = 4],y(t), singsol=all)
 

\[ y = -\frac {19 \sqrt {17}\, \sinh \left (\frac {\left (\pi -t \right ) \sqrt {17}}{2}\right ) {\mathrm e}^{-\frac {3 t}{2}+\frac {3 \pi }{2}}}{34}-\frac {\cosh \left (\frac {\left (\pi -t \right ) \sqrt {17}}{2}\right ) {\mathrm e}^{-\frac {3 t}{2}+\frac {3 \pi }{2}}}{2}+\frac {3 \,{\mathrm e}^{\pi -t}}{2} \]

✓ Solution by Mathematica

Time used: 0.339 (sec). Leaf size: 103

DSolve[{D[y[t],{t,2}]+3*D[y[t],t]-2*y[t]==-6*Exp[Pi-t],{y[Pi]==1,Derivative[1][y][Pi]==4}},y[t],t,IncludeSingularSolutions -> True]
 

\[ y(t)\to \frac {1}{68} e^{-\frac {1}{2} \left (3+\sqrt {17}\right ) t-\frac {1}{2} \left (\sqrt {17}-3\right ) \pi } \left (\left (19 \sqrt {17}-17\right ) e^{\sqrt {17} t}+102 e^{\frac {1}{2} \left (\left (1+\sqrt {17}\right ) t+\left (\sqrt {17}-1\right ) \pi \right )}-\left (\left (17+19 \sqrt {17}\right ) e^{\sqrt {17} \pi }\right )\right ) \]