83.21.4 problem 4

Internal problem ID [22064]
Book : Differential Equations By Kaj L. Nielsen. Second edition 1966. Barnes and nobel. 66-28306
Section : Examination VIII. page 256
Problem number : 4
Date solved : Thursday, October 02, 2025 at 08:23:19 PM
CAS classification : [[_3rd_order, _missing_x]]

\begin{align*} y^{\prime \prime \prime }-y^{\prime \prime }-6 y^{\prime }&=6 \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: 0.040 (sec). Leaf size: 21

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

\[ y = -t +\frac {2 \,{\mathrm e}^{3 t}}{15}+\frac {1}{6}-\frac {3 \,{\mathrm e}^{-2 t}}{10} \]

✓ Mathematica. Time used: 0.027 (sec). Leaf size: 28

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

\begin{align*} y(t)&\to \frac {1}{30} \left (-30 t-9 e^{-2 t}+4 e^{3 t}+5\right ) \end{align*}

✓ Sympy. Time used: 0.140 (sec). Leaf size: 24

from sympy import * 
t = symbols("t") 
y = Function("y") 
ode = Eq(-6*Derivative(y(t), t) - Derivative(y(t), (t, 2)) + Derivative(y(t), (t, 3)) - 6,0) 
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)
 

\[ y{\left (t \right )} = - t + \frac {2 e^{3 t}}{15} + \frac {1}{6} - \frac {3 e^{- 2 t}}{10} \]