76.18.6 problem 17

Internal problem ID [17717]
Book : Differential equations. An introduction to modern methods and applications. James Brannan, William E. Boyce. Third edition. Wiley 2015
Section : Chapter 5. The Laplace transform. Section 5.2 (Properties of the Laplace transform). Problems at page 309
Problem number : 17
Date solved : Tuesday, January 28, 2025 at 11:01:46 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} y^{\prime \prime }-5 y^{\prime }-6 y&=t^{2}+7 \end{align*}

Using Laplace method With initial conditions

\begin{align*} y \left (0\right )&=1\\ y^{\prime }\left (0\right )&=0 \end{align*}

✓ Solution by Maple

Time used: 9.788 (sec). Leaf size: 26

dsolve([diff(y(t),t$2)-5*diff(y(t),t)-6*y(t)=t^2+7,y(0) = 1, D(y)(0) = 0],y(t), singsol=all)
 

\[ y = -\frac {t^{2}}{6}+\frac {5 t}{18}+\frac {15 \,{\mathrm e}^{-t}}{7}-\frac {157}{108}+\frac {235 \,{\mathrm e}^{6 t}}{756} \]

✓ Solution by Mathematica

Time used: 0.015 (sec). Leaf size: 38

DSolve[{D[y[t],{t,2}]-5*D[y[t],t]-6*y[t]==t^2+7,{y[0]==1,Derivative[1][y][0] == 0}},y[t],t,IncludeSingularSolutions -> True]
 

\[ y(t)\to \frac {1}{756} e^{-t} \left (-7 e^t \left (18 t^2-30 t+157\right )+235 e^{7 t}+1620\right ) \]