67.12.22 problem 19.4 (f)

Internal problem ID [16661]
Book : Ordinary Differential Equations. An introduction to the fundamentals. Kenneth B. Howell. second edition. CRC Press. FL, USA. 2020
Section : Chapter 19. Arbitrary Homogeneous linear equations with constant coefficients. Additional exercises page 369
Problem number : 19.4 (f)
Date solved : Thursday, October 02, 2025 at 01:37:25 PM
CAS classification : [[_high_order, _missing_x]]

\begin{align*} y^{\left (6\right )}-2 y^{\prime \prime \prime }+y&=0 \end{align*}

✓ Maple. Time used: 0.002 (sec). Leaf size: 47

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

\[ y = {\mathrm e}^{-\frac {x}{2}} \left (c_6 x +c_4 \right ) \cos \left (\frac {\sqrt {3}\, x}{2}\right )+{\mathrm e}^{-\frac {x}{2}} \left (c_5 x +c_3 \right ) \sin \left (\frac {\sqrt {3}\, x}{2}\right )+{\mathrm e}^{x} \left (c_2 x +c_1 \right ) \]

✓ Mathematica. Time used: 0.003 (sec). Leaf size: 67

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

\begin{align*} y(x)&\to e^{-x/2} \left (e^{3 x/2} (c_6 x+c_5)+(c_4 x+c_3) \cos \left (\frac {\sqrt {3} x}{2}\right )+(c_2 x+c_1) \sin \left (\frac {\sqrt {3} x}{2}\right )\right ) \end{align*}

✓ Sympy. Time used: 0.109 (sec). Leaf size: 46

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

\[ y{\left (x \right )} = \left (C_{1} + C_{2} x\right ) e^{x} + \left (\left (C_{3} + C_{4} x\right ) \sin {\left (\frac {\sqrt {3} x}{2} \right )} + \left (C_{5} + C_{6} x\right ) \cos {\left (\frac {\sqrt {3} x}{2} \right )}\right ) e^{- \frac {x}{2}} \]