7.10.34 problem 34

Internal problem ID [304]
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 : 34
Date solved : Tuesday, March 04, 2025 at 11:07:33 AM
CAS classification : [[_3rd_order, _missing_x]]

\begin{align*} 3 y^{\prime \prime \prime }-2 y^{\prime \prime }+12 y^{\prime }-8 y&=0 \end{align*}

Using reduction of order method given that one solution is

\begin{align*} y&={\mathrm e}^{\frac {2 x}{3}} \end{align*}

✓ Maple. Time used: 0.003 (sec). Leaf size: 23

ode:=3*diff(diff(diff(y(x),x),x),x)-2*diff(diff(y(x),x),x)+12*diff(y(x),x)-8*y(x) = 0; 
dsolve(ode,y(x), singsol=all);
 

\[ y = c_1 \,{\mathrm e}^{\frac {2 x}{3}}+\sin \left (2 x \right ) c_2 +\cos \left (2 x \right ) c_3 \]

✓ Mathematica. Time used: 0.003 (sec). Leaf size: 30

ode=3*D[y[x],{x,3}]-2*D[y[x],{x,2}]+12*D[y[x],x]-8*y[x]==0; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\[ y(x)\to c_3 e^{2 x/3}+c_1 \cos (2 x)+c_2 \sin (2 x) \]

✓ Sympy. Time used: 0.180 (sec). Leaf size: 24

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(-8*y(x) + 12*Derivative(y(x), x) - 2*Derivative(y(x), (x, 2)) + 3*Derivative(y(x), (x, 3)),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

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