52.5.10 problem 40

Internal problem ID [8333]
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 : 40
Date solved : Wednesday, March 05, 2025 at 05:35:21 AM
CAS classification : [[_3rd_order, _linear, _nonhomogeneous]]

\begin{align*} y^{\prime \prime \prime }+2 y^{\prime \prime }-y^{\prime }-2 y&=\sin \left (3 t \right ) \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: 0.638 (sec). Leaf size: 31

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

\[ y = -\frac {13 \cosh \left (t \right )}{30}+\frac {13 \sinh \left (t \right )}{15}+\frac {16 \,{\mathrm e}^{-2 t}}{39}+\frac {3 \cos \left (3 t \right )}{130}-\frac {\sin \left (3 t \right )}{65} \]

✓ Mathematica. Time used: 0.067 (sec). Leaf size: 42

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

\[ y(t)\to \frac {1}{780} \left (e^{-2 t} \left (-507 e^t+169 e^{3 t}+320\right )-12 \sin (3 t)+18 \cos (3 t)\right ) \]

✓ Sympy. Time used: 0.285 (sec). Leaf size: 39

from sympy import * 
t = symbols("t") 
y = Function("y") 
ode = Eq(-2*y(t) - sin(3*t) - Derivative(y(t), t) + 2*Derivative(y(t), (t, 2)) + Derivative(y(t), (t, 3)),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)
 

\[ y{\left (t \right )} = \frac {13 e^{t}}{60} - \frac {\sin {\left (3 t \right )}}{65} + \frac {3 \cos {\left (3 t \right )}}{130} - \frac {13 e^{- t}}{20} + \frac {16 e^{- 2 t}}{39} \]