89.13.18 problem 18

Internal problem ID [24601]
Book : A short course in Differential Equations. Earl D. Rainville. Second edition. 1958. Macmillan Publisher, NY. CAT 58-5010
Section : Chapter 8. Linear Differential Equations with constant coefficients. Exercises at page 127
Problem number : 18
Date solved : Thursday, October 02, 2025 at 10:46:25 PM
CAS classification : [[_3rd_order, _missing_x]]

\begin{align*} 2 y^{\prime \prime \prime }-y^{\prime \prime }+36 y^{\prime }-18 y&=0 \end{align*}

✓ Maple. Time used: 0.001 (sec). Leaf size: 29

ode:=2*diff(diff(diff(y(x),x),x),x)-diff(diff(y(x),x),x)+36*diff(y(x),x)-18*y(x) = 0; 
dsolve(ode,y(x), singsol=all);
 

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

✓ Mathematica. Time used: 0.002 (sec). Leaf size: 40

ode=2*D[y[x],{x,3}]-D[y[x],{x,2}]+36*D[y[x],{x,1}]-18*y[x] ==0; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)&\to c_3 e^{x/2}+c_1 \cos \left (3 \sqrt {2} x\right )+c_2 \sin \left (3 \sqrt {2} x\right ) \end{align*}

✓ Sympy. Time used: 0.096 (sec). Leaf size: 32

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(-18*y(x) + 36*Derivative(y(x), x) - Derivative(y(x), (x, 2)) + 2*Derivative(y(x), (x, 3)),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

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