1.19.5 problem 31

Internal problem ID [545]
Book : Elementary Differential Equations. By C. Henry Edwards, David E. Penney and David Calvis. 6th edition. 2008
Section : Chapter 4. Laplace transform methods. Section 4.3 (Translation and partial fractions). Problems at page 296
Problem number : 31
Date solved : Tuesday, September 30, 2025 at 04:01:11 AM
CAS classification : [[_3rd_order, _missing_x]]

\begin{align*} x^{\prime \prime \prime }+x^{\prime \prime }-6 x^{\prime }&=0 \end{align*}

Using Laplace method With initial conditions

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

✓ Maple. Time used: 0.100 (sec). Leaf size: 18

ode:=diff(diff(diff(x(t),t),t),t)+diff(diff(x(t),t),t)-6*diff(x(t),t) = 0; 
ic:=[x(0) = 0, D(x)(0) = 1, (D@@2)(x)(0) = 1]; 
dsolve([ode,op(ic)],x(t),method='laplace');
 

\[ x = -\frac {1}{3}-\frac {{\mathrm e}^{-3 t}}{15}+\frac {2 \,{\mathrm e}^{2 t}}{5} \]

✓ Mathematica. Time used: 0.021 (sec). Leaf size: 25

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

\begin{align*} x(t)&\to \frac {1}{15} \left (-e^{-3 t}+6 e^{2 t}-5\right ) \end{align*}

✓ Sympy. Time used: 0.126 (sec). Leaf size: 22

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

\[ x{\left (t \right )} = \frac {2 e^{2 t}}{5} - \frac {1}{3} - \frac {e^{- 3 t}}{15} \]