64.10.23 problem 23

Internal problem ID [13358]
Book : Differential Equations by Shepley L. Ross. Third edition. John Willey. New Delhi. 2004.
Section : Chapter 4, Section 4.2. The homogeneous linear equation with constant coefficients. Exercises page 135
Problem number : 23
Date solved : Monday, March 31, 2025 at 07:52:17 AM
CAS classification : [[_high_order, _missing_x]]

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

✓ Maple. Time used: 0.004 (sec). Leaf size: 53

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

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

✓ Mathematica. Time used: 0.005 (sec). Leaf size: 65

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

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

✓ Sympy. Time used: 0.149 (sec). Leaf size: 70

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

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