66.17.14 problem 14

Internal problem ID [16237]
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 : 14
Date solved : Thursday, October 02, 2025 at 10:43:58 AM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} y^{\prime \prime }+2 y^{\prime }+y&=2 \cos \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: 0.021 (sec). Leaf size: 28

ode:=diff(diff(y(t),t),t)+2*diff(y(t),t)+y(t) = 2*cos(2*t); 
ic:=[y(0) = 0, D(y)(0) = 0]; 
dsolve([ode,op(ic)],y(t), singsol=all);
 

\[ y = \frac {2 \left (3-5 t \right ) {\mathrm e}^{-t}}{25}-\frac {6 \cos \left (2 t \right )}{25}+\frac {8 \sin \left (2 t \right )}{25} \]

✓ Mathematica. Time used: 0.015 (sec). Leaf size: 37

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

\begin{align*} y(t)&\to -\frac {2}{25} e^{-t} \left (5 t-4 e^t \sin (2 t)+3 e^t \cos (2 t)-3\right ) \end{align*}

✓ Sympy. Time used: 0.148 (sec). Leaf size: 31

from sympy import * 
t = symbols("t") 
y = Function("y") 
ode = Eq(y(t) - 2*cos(2*t) + 2*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)
 

\[ y{\left (t \right )} = \left (\frac {6}{25} - \frac {2 t}{5}\right ) e^{- t} + \frac {8 \sin {\left (2 t \right )}}{25} - \frac {6 \cos {\left (2 t \right )}}{25} \]