72.16.16 problem 16

Internal problem ID [14912]
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 : 16
Date solved : Tuesday, January 28, 2025 at 07:19:46 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} y^{\prime \prime }+4 y^{\prime }+20 y&={\mathrm e}^{-\frac {t}{2}} \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: 0.029 (sec). Leaf size: 30

dsolve([diff(y(t),t$2)+4*diff(y(t),t)+20*y(t)=exp(-t/2),y(0) = 0, D(y)(0) = 0],y(t), singsol=all)
 

\[ y = \frac {4 \,{\mathrm e}^{-\frac {t}{2}}}{73}+\frac {\left (-3 \sin \left (4 t \right )-8 \cos \left (4 t \right )\right ) {\mathrm e}^{-2 t}}{146} \]

✓ Solution by Mathematica

Time used: 0.168 (sec). Leaf size: 127

DSolve[{D[y[t],{t,2}]+4*D[y[t],t]+20*y[t]==Exp[-t/2],{y[0]==0,Derivative[1][y][0] ==0}},y[t],t,IncludeSingularSolutions -> True]
 

\[ y(t)\to e^{-2 t} \left (-\sin (4 t) \int _1^0\frac {1}{4} e^{\frac {3 K[1]}{2}} \cos (4 K[1])dK[1]+\sin (4 t) \int _1^t\frac {1}{4} e^{\frac {3 K[1]}{2}} \cos (4 K[1])dK[1]+\cos (4 t) \left (\int _1^t-\frac {1}{4} e^{\frac {3 K[2]}{2}} \sin (4 K[2])dK[2]-\int _1^0-\frac {1}{4} e^{\frac {3 K[2]}{2}} \sin (4 K[2])dK[2]\right )\right ) \]