52.5.6 problem 36

Internal problem ID [8329]
Book : DIFFERENTIAL EQUATIONS with Boundary Value Problems. DENNIS G. ZILL, WARREN S. WRIGHT, MICHAEL R. CULLEN. Brooks/Cole. Boston, MA. 2013. 8th edition.
Section : CHAPTER 7 THE LAPLACE TRANSFORM. 7.2.2 TRANSFORMS OF DERIVATIVES Page 289
Problem number : 36
Date solved : Wednesday, March 05, 2025 at 05:35:17 AM
CAS classification : [[_2nd_order, _missing_y]]

\begin{align*} y^{\prime \prime }-4 y^{\prime }&=6 \,{\mathrm e}^{3 t}-3 \,{\mathrm e}^{-t} \end{align*}

Using Laplace method With initial conditions

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

✓ Maple. Time used: 0.686 (sec). Leaf size: 24

ode:=diff(diff(y(t),t),t)-4*diff(y(t),t) = 6*exp(3*t)-3*exp(-t); 
ic:=y(0) = 1, D(y)(0) = -1; 
dsolve([ode,ic],y(t),method='laplace');
 

\[ y = \frac {11 \,{\mathrm e}^{4 t}}{10}-2 \,{\mathrm e}^{3 t}-\frac {3 \,{\mathrm e}^{-t}}{5}+\frac {5}{2} \]

✓ Mathematica. Time used: 0.158 (sec). Leaf size: 34

ode=D[y[t],{t,2}]-4*D[y[t],t]==6*Exp[3*t]-3*Exp[-t]; 
ic={y[0]==1,Derivative[1][y][0] ==-1}; 
DSolve[{ode,ic},y[t],t,IncludeSingularSolutions->True]
 

\[ y(t)\to -\frac {3 e^{-t}}{5}-2 e^{3 t}+\frac {11 e^{4 t}}{10}+\frac {5}{2} \]

✓ Sympy. Time used: 0.266 (sec). Leaf size: 27

from sympy import * 
t = symbols("t") 
y = Function("y") 
ode = Eq(-6*exp(3*t) - 4*Derivative(y(t), t) + Derivative(y(t), (t, 2)) + 3*exp(-t),0) 
ics = {y(0): 1, Subs(Derivative(y(t), t), t, 0): -1} 
dsolve(ode,func=y(t),ics=ics)
 

\[ y{\left (t \right )} = \frac {11 e^{4 t}}{10} - 2 e^{3 t} + \frac {5}{2} - \frac {3 e^{- t}}{5} \]