82.29.2 problem Ex. 3

Internal problem ID [18813]
Book : Introductory Course On Differential Equations by Daniel A Murray. Longmans Green and Co. NY. 1924
Section : Chapter VI. Linear equations with constant coefficients. problems at page 76
Problem number : Ex. 3
Date solved : Monday, March 31, 2025 at 06:15:33 PM
CAS classification : [[_3rd_order, _linear, _nonhomogeneous]]

\begin{align*} y^{\prime \prime \prime }+8 y&=x^{4}+2 x +1 \end{align*}

✓ Maple. Time used: 0.003 (sec). Leaf size: 40

ode:=diff(diff(diff(y(x),x),x),x)+8*y(x) = x^4+2*x+1; 
dsolve(ode,y(x), singsol=all);
 

\[ y = \frac {x^{4}}{8}-\frac {x}{8}+\frac {1}{8}+c_1 \,{\mathrm e}^{-2 x}+c_2 \,{\mathrm e}^{x} \cos \left (\sqrt {3}\, x \right )+c_3 \,{\mathrm e}^{x} \sin \left (\sqrt {3}\, x \right ) \]

✓ Mathematica. Time used: 0.004 (sec). Leaf size: 54

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

\[ y(x)\to \frac {1}{8} \left (x^4-x+1\right )+c_1 e^{-2 x}+c_3 e^x \cos \left (\sqrt {3} x\right )+c_2 e^x \sin \left (\sqrt {3} x\right ) \]

✓ Sympy. Time used: 0.171 (sec). Leaf size: 44

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(-x**4 - 2*x + 8*y(x) + Derivative(y(x), (x, 3)) - 1,0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

\[ y{\left (x \right )} = C_{3} e^{- 2 x} + \frac {x^{4}}{8} - \frac {x}{8} + \left (C_{1} \sin {\left (\sqrt {3} x \right )} + C_{2} \cos {\left (\sqrt {3} x \right )}\right ) e^{x} + \frac {1}{8} \]