72.16.38 problem 40

Internal problem ID [14934]
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 : 40
Date solved : Tuesday, January 28, 2025 at 07:21:09 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: 0.019 (sec). Leaf size: 37

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)
 

\[ y = \frac {\left (16 \,{\mathrm e}^{5 t}+60 \,{\mathrm e}^{4 t} t -45 \,{\mathrm e}^{4 t}+20 \,{\mathrm e}^{2 t}+9\right ) {\mathrm e}^{-4 t}}{240} \]

✓ Solution by Mathematica

Time used: 0.342 (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]
 

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