72.16.37 problem 39

Internal problem ID [14933]
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 : 39
Date solved : Tuesday, January 28, 2025 at 07:21:07 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} y^{\prime \prime }+6 y^{\prime }+8 y&=2 t +{\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: 0.015 (sec). Leaf size: 27

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

\[ y = \frac {5 \,{\mathrm e}^{-4 t}}{48}-\frac {3}{16}+\frac {t}{4}+\frac {{\mathrm e}^{-t}}{3}-\frac {{\mathrm e}^{-2 t}}{4} \]

✓ Solution by Mathematica

Time used: 0.237 (sec). Leaf size: 127

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

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