72.16.15 problem 15

Internal problem ID [14911]
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 : 15
Date solved : Tuesday, January 28, 2025 at 07:19:44 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

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

\[ y = -\frac {{\mathrm e}^{-3 t}}{2}+\frac {{\mathrm e}^{-t}}{6}+\frac {{\mathrm e}^{-4 t}}{3} \]

✓ Solution by Mathematica

Time used: 0.035 (sec). Leaf size: 26

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

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