82.33.15 problem Ex. 15

Internal problem ID [18829]
Book : Introductory Course On Differential Equations by Daniel A Murray. Longmans Green and Co. NY. 1924
Section : Chapter VI. Linear equations with constant coefficients. Examples on chapter VI, page 80
Problem number : Ex. 15
Date solved : Thursday, March 13, 2025 at 01:00:18 PM
CAS classification : [[_high_order, _linear, _nonhomogeneous]]

\begin{align*} y^{\prime \prime \prime \prime }-a^{4} y&=x^{4} \end{align*}

✓ Maple. Time used: 0.008 (sec). Leaf size: 55

ode:=diff(diff(diff(diff(y(x),x),x),x),x)-a^4*y(x) = x^4; 
dsolve(ode,y(x), singsol=all);
 

\[ y \left (x \right ) = \frac {c_3 \sin \left (a x \right ) a^{8}+\cos \left (a x \right ) c_{1} a^{8}+c_{2} {\mathrm e}^{a x} a^{8}+c_4 \,{\mathrm e}^{-a x} a^{8}-x^{4} a^{4}-24}{a^{8}} \]

✓ Mathematica. Time used: 0.004 (sec). Leaf size: 51

ode=D[y[x],{x,4}]-a^4*y[x]==x^4; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\[ y(x)\to -\frac {a^4 x^4+24}{a^8}+c_2 e^{-a x}+c_4 e^{a x}+c_1 \cos (a x)+c_3 \sin (a x) \]

✓ Sympy. Time used: 0.170 (sec). Leaf size: 44

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

\[ y{\left (x \right )} = C_{1} e^{- a x} + C_{2} e^{a x} + C_{3} e^{- i a x} + C_{4} e^{i a x} - \frac {x^{4}}{a^{4}} - \frac {24}{a^{8}} \]