71.10.5 problem 5

Internal problem ID [14429]
Book : Ordinary Differential Equations by Charles E. Roberts, Jr. CRC Press. 2010
Section : Chapter 4. N-th Order Linear Differential Equations. Exercises 4.3, page 210
Problem number : 5
Date solved : Monday, March 31, 2025 at 12:26:59 PM
CAS classification : [[_high_order, _missing_x]]

\begin{align*} y^{\prime \prime \prime \prime }-4 y^{\prime \prime \prime }+8 y^{\prime \prime }-8 y^{\prime }+4 y&=0 \end{align*}

✓ Maple. Time used: 0.002 (sec). Leaf size: 24

ode:=diff(diff(diff(diff(y(x),x),x),x),x)-4*diff(diff(diff(y(x),x),x),x)+8*diff(diff(y(x),x),x)-8*diff(y(x),x)+4*y(x) = 0; 
dsolve(ode,y(x), singsol=all);
 

\[ y = \left (\left (c_4 x +c_2 \right ) \cos \left (x \right )+\sin \left (x \right ) \left (c_3 x +c_1 \right )\right ) {\mathrm e}^{x} \]

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

ode=D[y[x],{x,4}]-4*D[y[x],{x,3}]+8*D[y[x],{x,2}]-8*D[y[x],x]+4*y[x]==0; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\[ y(x)\to e^x ((c_4 x+c_3) \cos (x)+(c_2 x+c_1) \sin (x)) \]

✓ Sympy. Time used: 0.198 (sec). Leaf size: 22

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(4*y(x) - 8*Derivative(y(x), x) + 8*Derivative(y(x), (x, 2)) - 4*Derivative(y(x), (x, 3)) + Derivative(y(x), (x, 4)),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

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