14.16.7 problem 23

Internal problem ID [2677]
Book : Differential equations and their applications, 4th ed., M. Braun
Section : Chapter 2. Second order differential equations. Section 2.9, The method of Laplace transform. Excercises page 232
Problem number : 23
Date solved : Sunday, March 30, 2025 at 12:13:52 AM
CAS classification : [[_3rd_order, _with_linear_symmetries]]

\begin{align*} y^{\prime \prime \prime }-6 y^{\prime \prime }+11 y^{\prime }-6 y&={\mathrm e}^{4 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 )&=0 \end{align*}

✓ Maple. Time used: 0.107 (sec). Leaf size: 14

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

\[ y = \frac {{\mathrm e}^{t} \left ({\mathrm e}^{t}-1\right )^{3}}{6} \]

✓ Mathematica. Time used: 0.013 (sec). Leaf size: 19

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

\[ y(t)\to \frac {1}{6} e^t \left (e^t-1\right )^3 \]

✓ Sympy. Time used: 0.279 (sec). Leaf size: 27

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

\[ y{\left (t \right )} = \left (\frac {e^{3 t}}{6} - \frac {e^{2 t}}{2} + \frac {e^{t}}{2} - \frac {1}{6}\right ) e^{t} \]