72.16.9 problem 9

Internal problem ID [14905]
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 : 9
Date solved : Tuesday, January 28, 2025 at 07:19:31 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} y^{\prime \prime }+6 y^{\prime }+8 y&={\mathrm e}^{-t} \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: 0.018 (sec). Leaf size: 24

dsolve([diff(y(t),t$2)+6*diff(y(t),t)+8*y(t)=exp(-t),y(0) = 0, D(y)(0) = 0],y(t), singsol=all)
 

\[ y = -\frac {\left (-2 \,{\mathrm e}^{3 t}+3 \,{\mathrm e}^{2 t}-1\right ) {\mathrm e}^{-4 t}}{6} \]

✓ Solution by Mathematica

Time used: 0.030 (sec). Leaf size: 28

DSolve[{D[y[t],{t,2}]+6*D[y[t],t]+8*y[t]==Exp[-t],{y[0]==0,Derivative[1][y][0] ==0}},y[t],t,IncludeSingularSolutions -> True]
 

\[ y(t)\to \frac {1}{6} e^{-4 t} \left (e^t-1\right )^2 \left (2 e^t+1\right ) \]