15.10.18 problem 18

Internal problem ID [3105]
Book : Differential Equations by Alfred L. Nelson, Karl W. Folley, Max Coral. 3rd ed. DC heath. Boston. 1964
Section : Exercise 18, page 82
Problem number : 18
Date solved : Sunday, March 30, 2025 at 01:18:33 AM
CAS classification : [[_high_order, _missing_x]]

\begin{align*} y^{\prime \prime \prime \prime }+y^{\prime \prime \prime }-3 y^{\prime \prime }-4 y^{\prime }-4 y&=0 \end{align*}

✓ Maple. Time used: 0.002 (sec). Leaf size: 43

ode:=diff(diff(diff(diff(y(x),x),x),x),x)+diff(diff(diff(y(x),x),x),x)-3*diff(diff(y(x),x),x)-4*diff(y(x),x)-4*y(x) = 0; 
dsolve(ode,y(x), singsol=all);
 

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

✓ Mathematica. Time used: 0.004 (sec). Leaf size: 64

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

\[ y(x)\to e^{-2 x} \left (c_4 e^{4 x}+c_2 e^{3 x/2} \cos \left (\frac {\sqrt {3} x}{2}\right )+c_1 e^{3 x/2} \sin \left (\frac {\sqrt {3} x}{2}\right )+c_3\right ) \]

✓ Sympy. Time used: 0.295 (sec). Leaf size: 44

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

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