72.17.13 problem 13

Internal problem ID [14870]
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 4.2 page 412
Problem number : 13
Date solved : Thursday, March 13, 2025 at 05:15:57 AM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} y^{\prime \prime }+6 y^{\prime }+20 y&=-3 \sin \left (2 t \right ) \end{align*}

With initial conditions

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

✓ Maple. Time used: 0.045 (sec). Leaf size: 44

ode:=diff(diff(y(t),t),t)+6*diff(y(t),t)+20*y(t) = -3*sin(2*t); 
ic:=y(0) = 0, D(y)(0) = 0; 
dsolve([ode,ic],y(t), singsol=all);
 

\[ y = -\frac {3 \,{\mathrm e}^{-3 t} \sqrt {11}\, \sin \left (\sqrt {11}\, t \right )}{1100}-\frac {9 \,{\mathrm e}^{-3 t} \cos \left (\sqrt {11}\, t \right )}{100}-\frac {3 \sin \left (2 t \right )}{25}+\frac {9 \cos \left (2 t \right )}{100} \]

✓ Mathematica. Time used: 0.026 (sec). Leaf size: 61

ode=D[y[t],{t,2}]+6*D[y[t],t]+20*y[t]==-3*Sin[2*t]; 
ic={y[0]==0,Derivative[1][y][0] ==0}; 
DSolve[{ode,ic},y[t],t,IncludeSingularSolutions->True]
 

\[ y(t)\to -\frac {3 e^{-3 t} \left (44 e^{3 t} \sin (2 t)+\sqrt {11} \sin \left (\sqrt {11} t\right )-33 e^{3 t} \cos (2 t)+33 \cos \left (\sqrt {11} t\right )\right )}{1100} \]

✓ Sympy. Time used: 0.270 (sec). Leaf size: 54

from sympy import * 
t = symbols("t") 
y = Function("y") 
ode = Eq(20*y(t) + 3*sin(2*t) + 6*Derivative(y(t), t) + Derivative(y(t), (t, 2)),0) 
ics = {y(0): 0, Subs(Derivative(y(t), t), t, 0): 0} 
dsolve(ode,func=y(t),ics=ics)
 

\[ y{\left (t \right )} = \left (- \frac {3 \sqrt {11} \sin {\left (\sqrt {11} t \right )}}{1100} - \frac {9 \cos {\left (\sqrt {11} t \right )}}{100}\right ) e^{- 3 t} - \frac {3 \sin {\left (2 t \right )}}{25} + \frac {9 \cos {\left (2 t \right )}}{100} \]