89.14.20 problem 20

Internal problem ID [24624]
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 128
Problem number : 20
Date solved : Thursday, October 02, 2025 at 10:46:33 PM
CAS classification : [[_high_order, _missing_x]]

\begin{align*} y^{\prime \prime \prime \prime }+5 y^{\prime \prime }+4 y&=0 \end{align*}

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

ode:=diff(diff(diff(diff(y(x),x),x),x),x)+5*diff(diff(y(x),x),x)+4*y(x) = 0; 
dsolve(ode,y(x), singsol=all);
 

\[ y = 2 c_2 \cos \left (x \right )^{2}+\left (2 c_1 \sin \left (x \right )+c_4 \right ) \cos \left (x \right )+c_3 \sin \left (x \right )-c_2 \]

✓ Mathematica. Time used: 0.002 (sec). Leaf size: 30

ode=D[y[x],{x,4}]+5*D[y[x],{x,2}]+4*y[x]==0; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)&\to c_1 \cos (2 x)+c_4 \sin (x)+\cos (x) (2 c_2 \sin (x)+c_3) \end{align*}

✓ Sympy. Time used: 0.050 (sec). Leaf size: 26

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(4*y(x) + 5*Derivative(y(x), (x, 2)) + Derivative(y(x), (x, 4)),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

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