72.16.23 problem 24

Internal problem ID [14840]
Book : DIFFERENTIAL EQUATIONS by Paul Blanchard, Robert L. Devaney, Glen R. Hall. 4th edition. Brooks/Cole. Boston, USA. 2012
Section : Chapter 4. Forcing and Resonance. Section 4.1 page 399
Problem number : 24
Date solved : Thursday, March 13, 2025 at 04:21:10 AM
CAS classification : [[_2nd_order, _missing_x]]

\begin{align*} y^{\prime \prime }+4 y^{\prime }+6 y&=-8 \end{align*}

With initial conditions

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

✓ Maple. Time used: 0.038 (sec). Leaf size: 33

ode:=diff(diff(y(t),t),t)+4*diff(y(t),t)+6*y(t) = -8; 
ic:=y(0) = 0, D(y)(0) = 0; 
dsolve([ode,ic],y(t), singsol=all);
 

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

✓ Mathematica. Time used: 0.023 (sec). Leaf size: 44

ode=D[y[t],{t,2}]+4*D[y[t],t]+6*y[t]==-8; 
ic={y[0]==0,Derivative[1][y][0] ==0}; 
DSolve[{ode,ic},y[t],t,IncludeSingularSolutions->True]
 

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

✓ Sympy. Time used: 0.223 (sec). Leaf size: 39

from sympy import * 
t = symbols("t") 
y = Function("y") 
ode = Eq(6*y(t) + 4*Derivative(y(t), t) + Derivative(y(t), (t, 2)) + 8,0) 
ics = {y(0): 0, Subs(Derivative(y(t), t), t, 0): 0} 
dsolve(ode,func=y(t),ics=ics)
 

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