7.10.33 problem 33

Internal problem ID [303]
Book : Elementary Differential Equations. By C. Henry Edwards, David E. Penney and David Calvis. 6th edition. 2008
Section : Chapter 2. Linear Equations of Higher Order. Section 2.3 (Homogeneous equations with constant coefficients). Problems at page 134
Problem number : 33
Date solved : Tuesday, March 04, 2025 at 11:07:32 AM
CAS classification : [[_3rd_order, _missing_x]]

\begin{align*} y^{\prime \prime \prime }+3 y^{\prime \prime }-54 y&=0 \end{align*}

Using reduction of order method given that one solution is

\begin{align*} y&={\mathrm e}^{3 x} \end{align*}

✓ Maple. Time used: 0.002 (sec). Leaf size: 31

ode:=diff(diff(diff(y(x),x),x),x)+3*diff(diff(y(x),x),x)-54*y(x) = 0; 
dsolve(ode,y(x), singsol=all);
 

\[ y = c_1 \,{\mathrm e}^{3 x}+c_2 \,{\mathrm e}^{-3 x} \sin \left (3 x \right )+c_3 \,{\mathrm e}^{-3 x} \cos \left (3 x \right ) \]

✓ Mathematica. Time used: 0.003 (sec). Leaf size: 34

ode=D[y[x],{x,3}]+3*D[y[x],{x,2}]-54*y[x]==0; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\[ y(x)\to e^{-3 x} \left (c_3 e^{6 x}+c_2 \cos (3 x)+c_1 \sin (3 x)\right ) \]

✓ Sympy. Time used: 0.115 (sec). Leaf size: 27

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(-54*y(x) + 3*Derivative(y(x), (x, 2)) + Derivative(y(x), (x, 3)),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

\[ y{\left (x \right )} = C_{3} e^{3 x} + \left (C_{1} \sin {\left (3 x \right )} + C_{2} \cos {\left (3 x \right )}\right ) e^{- 3 x} \]