64.20.12 problem 12

Internal problem ID [13576]
Book : Differential Equations by Shepley L. Ross. Third edition. John Willey. New Delhi. 2004.
Section : Chapter 9, The Laplace transform. Section 9.3, Exercises page 452
Problem number : 12
Date solved : Wednesday, March 05, 2025 at 10:04:01 PM
CAS classification : [[_3rd_order, _linear, _nonhomogeneous]]

\begin{align*} y^{\prime \prime \prime }-6 y^{\prime \prime }+11 y^{\prime }-6 y&=36 t \,{\mathrm e}^{4 t} \end{align*}

Using Laplace method With initial conditions

\begin{align*} y \left (0\right )&=-1\\ y^{\prime }\left (0\right )&=0\\ y^{\prime \prime }\left (0\right )&=-6 \end{align*}

✓ Maple. Time used: 8.418 (sec). Leaf size: 25

ode:=diff(diff(diff(y(t),t),t),t)-6*diff(diff(y(t),t),t)+11*diff(y(t),t)-6*y(t) = 36*t*exp(4*t); 
ic:=y(0) = -1, D(y)(0) = 0, (D@@2)(y)(0) = -6; 
dsolve([ode,ic],y(t),method='laplace');
 

\[ y = -4 \,{\mathrm e}^{t}+14 \,{\mathrm e}^{3 t}+{\mathrm e}^{4 t} \left (6 t -11\right ) \]

✓ Mathematica. Time used: 0.007 (sec). Leaf size: 29

ode=D[y[t],{t,3}]-6*D[y[t],{t,2}]+11*D[y[t],t]-6*y[t]==36*t*Exp[4*t]; 
ic={y[0]==-1,Derivative[1][y][0]==0,Derivative[2][y][0]==-6}; 
DSolve[{ode,ic},{y[t]},t,IncludeSingularSolutions->True]
 

\[ y(t)\to e^t \left (e^{3 t} (6 t-11)+14 e^{2 t}-4\right ) \]

✓ Sympy. Time used: 0.318 (sec). Leaf size: 24

from sympy import * 
t = symbols("t") 
y = Function("y") 
ode = Eq(-36*t*exp(4*t) - 6*y(t) + 11*Derivative(y(t), t) - 6*Derivative(y(t), (t, 2)) + Derivative(y(t), (t, 3)),0) 
ics = {y(0): -1, Subs(Derivative(y(t), t), t, 0): 0, Subs(Derivative(y(t), (t, 2)), t, 0): -6} 
dsolve(ode,func=y(t),ics=ics)
 

\[ y{\left (t \right )} = \left (\left (6 t - 11\right ) e^{3 t} + 14 e^{2 t} - 4\right ) e^{t} \]