44.25.6 problem 4(b)

Internal problem ID [9464]
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 : 4(b)
Date solved : Tuesday, September 30, 2025 at 06:19:05 PM
CAS classification : [[_2nd_order, _missing_x]]

\begin{align*} y^{\prime \prime }+3 y^{\prime }+3 y&=2 \end{align*}

Using Laplace method

✓ Maple. Time used: 0.158 (sec). Leaf size: 48

ode:=diff(diff(y(t),t),t)+3*diff(y(t),t)+3*y(t) = 2; 
dsolve(ode,y(t),method='laplace');
 

\[ y = \frac {2}{3}+\frac {\left (\cos \left (\frac {\sqrt {3}\, t}{2}\right ) \left (-2+3 y \left (0\right )\right )+\sin \left (\frac {\sqrt {3}\, t}{2}\right ) \sqrt {3}\, \left (2 y^{\prime }\left (0\right )+3 y \left (0\right )-2\right )\right ) {\mathrm e}^{-\frac {3 t}{2}}}{3} \]

✓ Mathematica. Time used: 0.013 (sec). Leaf size: 51

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

\begin{align*} y(t)&\to c_2 e^{-3 t/2} \cos \left (\frac {\sqrt {3} t}{2}\right )+c_1 e^{-3 t/2} \sin \left (\frac {\sqrt {3} t}{2}\right )+\frac {2}{3} \end{align*}

✓ Sympy. Time used: 0.122 (sec). Leaf size: 36

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

\[ y{\left (t \right )} = \left (C_{1} \sin {\left (\frac {\sqrt {3} t}{2} \right )} + C_{2} \cos {\left (\frac {\sqrt {3} t}{2} \right )}\right ) e^{- \frac {3 t}{2}} + \frac {2}{3} \]