89.12.20 problem 20

Internal problem ID [24574]
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 121
Problem number : 20
Date solved : Thursday, October 02, 2025 at 10:46:11 PM
CAS classification : [[_high_order, _missing_x]]

\begin{align*} y^{\left (5\right )}-5 y^{\prime \prime \prime \prime }+7 y^{\prime \prime \prime }+y^{\prime \prime }-8 y^{\prime }+4 y&=0 \end{align*}

✓ Maple. Time used: 0.002 (sec). Leaf size: 31

ode:=diff(diff(diff(diff(diff(y(x),x),x),x),x),x)-5*diff(diff(diff(diff(y(x),x),x),x),x)+7*diff(diff(diff(y(x),x),x),x)+diff(diff(y(x),x),x)-8*diff(y(x),x)+4*y(x) = 0; 
dsolve(ode,y(x), singsol=all);
 

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

✓ Mathematica. Time used: 0.003 (sec). Leaf size: 38

ode=D[y[x],{x,5}]-5*D[y[x],{x,4}]+7*D[y[x],{x,3}]+D[y[x],{x,2}]-8*D[y[x],{x,1}]+ 4*y[x] ==0; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)&\to c_1 e^{-x}+e^x (c_3 x+c_2)+e^{2 x} (c_5 x+c_4) \end{align*}

✓ Sympy. Time used: 0.152 (sec). Leaf size: 27

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(4*y(x) - 8*Derivative(y(x), x) + Derivative(y(x), (x, 2)) + 7*Derivative(y(x), (x, 3)) - 5*Derivative(y(x), (x, 4)) + Derivative(y(x), (x, 5)),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

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