44.25.2 problem 3(b)

Internal problem ID [9460]
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 : Tuesday, September 30, 2025 at 06:19:03 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*}

✓ Maple. Time used: 0.133 (sec). Leaf size: 53

ode:=diff(diff(y(t),t),t)+3*diff(y(t),t)-2*y(t) = -6*exp(Pi-t); 
ic:=[y(Pi) = 1, D(y)(Pi) = 4]; 
dsolve([ode,op(ic)],y(t),method='laplace');
 

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

✓ Mathematica. Time used: 0.202 (sec). Leaf size: 103

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

\begin{align*} 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 ) \end{align*}

✓ Sympy. Time used: 0.206 (sec). Leaf size: 124

from sympy import * 
t = symbols("t") 
y = Function("y") 
ode = Eq(-2*y(t) + 6*exp(pi - t) + 3*Derivative(y(t), t) + Derivative(y(t), (t, 2)),0) 
ics = {y(pi): 1, Subs(Derivative(y(t), t), t, pi): 4} 
dsolve(ode,func=y(t),ics=ics)
 

\[ y{\left (t \right )} = \left (- \frac {e^{\frac {3 \pi }{2}}}{4 e^{\frac {\sqrt {17} \pi }{2}}} + \frac {19 \sqrt {17} e^{\frac {3 \pi }{2}}}{68 e^{\frac {\sqrt {17} \pi }{2}}}\right ) e^{\frac {t \left (-3 + \sqrt {17}\right )}{2}} + \frac {3 e^{\pi - t}}{2} + \left (- \frac {19 \sqrt {17} e^{\frac {3 \pi }{2}} e^{\frac {\sqrt {17} \pi }{2}}}{68} - \frac {e^{\frac {3 \pi }{2}} e^{\frac {\sqrt {17} \pi }{2}}}{4}\right ) e^{- \frac {t \left (3 + \sqrt {17}\right )}{2}} \]