52.5.9 problem 39

Internal problem ID [8332]
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 : 39
Date solved : Wednesday, March 05, 2025 at 05:35:20 AM
CAS classification : [[_3rd_order, _with_linear_symmetries]]

\begin{align*} 2 y^{\prime \prime \prime }+3 y^{\prime \prime }-3 y^{\prime }-2 y&={\mathrm e}^{-t} \end{align*}

Using Laplace method With initial conditions

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

✓ Maple. Time used: 0.743 (sec). Leaf size: 25

ode:=2*diff(diff(diff(y(t),t),t),t)+3*diff(diff(y(t),t),t)-3*diff(y(t),t)-2*y(t) = exp(-t); 
ic:=y(0) = 0, D(y)(0) = 0, (D@@2)(y)(0) = 1; 
dsolve([ode,ic],y(t),method='laplace');
 

\[ y = \frac {7 \cosh \left (t \right )}{9}-\frac {2 \sinh \left (t \right )}{9}+\frac {{\mathrm e}^{-2 t}}{9}-\frac {8 \,{\mathrm e}^{-\frac {t}{2}}}{9} \]

✓ Mathematica. Time used: 0.007 (sec). Leaf size: 37

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

\[ y(t)\to \frac {1}{18} e^{-2 t} \left (9 e^t-16 e^{3 t/2}+5 e^{3 t}+2\right ) \]

✓ Sympy. Time used: 0.282 (sec). Leaf size: 32

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

\[ y{\left (t \right )} = \frac {5 e^{t}}{18} + \frac {e^{- t}}{2} + \frac {e^{- 2 t}}{9} - \frac {8 e^{- \frac {t}{2}}}{9} \]