1.10.35 problem 35

Internal problem ID [305]
Book : Elementary Differential Equations. By C. Henry Edwards, David E. Penney and David Calvis. 6th edition. 2008
Section : Chapter 2. Linear Equations of Higher Order. Section 2.3 (Homogeneous equations with constant coefficients). Problems at page 134
Problem number : 35
Date solved : Tuesday, September 30, 2025 at 03:54:43 AM
CAS classification : [[_high_order, _missing_x]]

\begin{align*} 6 y^{\prime \prime \prime \prime }+5 y^{\prime \prime \prime }+25 y^{\prime \prime }+20 y^{\prime }+4 y&=0 \end{align*}

Using reduction of order method given that one solution is

\begin{align*} y&=\cos \left (2 x \right ) \end{align*}

✓ Maple. Time used: 0.003 (sec). Leaf size: 29

ode:=6*diff(diff(diff(diff(y(x),x),x),x),x)+5*diff(diff(diff(y(x),x),x),x)+25*diff(diff(y(x),x),x)+20*diff(y(x),x)+4*y(x) = 0; 
dsolve(ode,y(x), singsol=all);
 

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

✓ Mathematica. Time used: 0.002 (sec). Leaf size: 41

ode=6*D[y[x],{x,4}]+5*D[y[x],{x,3}]+25*D[y[x],{x,2}]+20*D[y[x],x]+4*y[x]==0; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)&\to e^{-x/2} \left (c_3 e^{x/6}+c_4\right )+c_1 \cos (2 x)+c_2 \sin (2 x) \end{align*}

✓ Sympy. Time used: 0.120 (sec). Leaf size: 29

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(4*y(x) + 20*Derivative(y(x), x) + 25*Derivative(y(x), (x, 2)) + 5*Derivative(y(x), (x, 3)) + 6*Derivative(y(x), (x, 4)),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

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