76.18.6 problem 17

Internal problem ID [17646]
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 : Monday, March 31, 2025 at 04:23:32 PM
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*}

✓ Maple. Time used: 0.101 (sec). Leaf size: 26

ode:=diff(diff(y(t),t),t)-5*diff(y(t),t)-6*y(t) = t^2+7; 
ic:=y(0) = 1, D(y)(0) = 0; 
dsolve([ode,ic],y(t),method='laplace');
 

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

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

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

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

✓ Sympy. Time used: 0.218 (sec). Leaf size: 31

from sympy import * 
t = symbols("t") 
y = Function("y") 
ode = Eq(-t**2 - 6*y(t) - 5*Derivative(y(t), t) + Derivative(y(t), (t, 2)) - 7,0) 
ics = {y(0): 1, Subs(Derivative(y(t), t), t, 0): 0} 
dsolve(ode,func=y(t),ics=ics)
 

\[ y{\left (t \right )} = - \frac {t^{2}}{6} + \frac {5 t}{18} + \frac {235 e^{6 t}}{756} - \frac {157}{108} + \frac {15 e^{- t}}{7} \]