72.17.14 problem 14

Internal problem ID [14950]
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 : Tuesday, January 28, 2025 at 07:24:34 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*}

✓ Solution by Maple

Time used: 0.023 (sec). Leaf size: 28

dsolve([diff(y(t),t$2)+2*diff(y(t),t)+y(t)=2*cos(2*t),y(0) = 0, D(y)(0) = 0],y(t), singsol=all)
 

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

✓ Solution by Mathematica

Time used: 0.019 (sec). Leaf size: 37

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

\[ y(t)\to -\frac {2}{25} e^{-t} \left (5 t-4 e^t \sin (2 t)+3 e^t \cos (2 t)-3\right ) \]