78.26.2 problem 4 (b)

Internal problem ID [18463]
Book : DIFFERENTIAL EQUATIONS WITH APPLICATIONS AND HISTORICAL NOTES by George F. Simmons. 3rd edition. 2017. CRC press, Boca Raton FL.
Section : Chapter 9. Laplace transforms. Section 53. More about Convolutions. The Unit Step and Impulse Functions. Problems at page 481
Problem number : 4 (b)
Date solved : Tuesday, January 28, 2025 at 11:50:26 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} y^{\prime \prime }+y^{\prime }-6 y&=t \end{align*}

Using Laplace method With initial conditions

\begin{align*} y \left (0\right )&=0\\ y^{\prime }\left (0\right )&=0 \end{align*}

✓ Solution by Maple

Time used: 0.158 (sec). Leaf size: 31

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

\[ y = -\frac {\left (-9 \,{\mathrm e}^{5 t}+30 t \,{\mathrm e}^{3 t}+5 \,{\mathrm e}^{3 t}+4\right ) {\mathrm e}^{-3 t}}{180} \]

✓ Solution by Mathematica

Time used: 0.015 (sec). Leaf size: 28

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

\[ y(t)\to \frac {1}{180} \left (-30 t-4 e^{-3 t}+9 e^{2 t}-5\right ) \]