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 : Wednesday, March 05, 2025 at 05:31:20 AM
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*}

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

ode:=diff(diff(y(t),t),t)-diff(y(t),t)+y(t) = 3*exp(-t); 
ic:=y(0) = 3, D(y)(0) = 2; 
dsolve([ode,ic],y(t),method='laplace');
 

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

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

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

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

✓ Sympy. Time used: 0.259 (sec). Leaf size: 42

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

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