72.16.13 problem 13

Internal problem ID [14909]
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 : 13
Date solved : Tuesday, January 28, 2025 at 07:19:41 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} y^{\prime \prime }+4 y^{\prime }+3 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: 0.017 (sec). Leaf size: 23

dsolve([diff(y(t),t$2)+4*diff(y(t),t)+3*y(t)=exp(-t/2),y(0) = 0, D(y)(0) = 0],y(t), singsol=all)
 

\[ y = \frac {{\mathrm e}^{-3 t}}{5}-{\mathrm e}^{-t}+\frac {4 \,{\mathrm e}^{-\frac {t}{2}}}{5} \]

✓ Solution by Mathematica

Time used: 0.051 (sec). Leaf size: 32

DSolve[{D[y[t],{t,2}]+4*D[y[t],t]+3*y[t]==Exp[-t/2],{y[0]==0,Derivative[1][y][0] ==0}},y[t],t,IncludeSingularSolutions -> True]
 

\[ y(t)\to \frac {1}{5} e^{-3 t} \left (-5 e^{2 t}+4 e^{5 t/2}+1\right ) \]