89.22.15 problem 15

Internal problem ID [24803]
Book : A short course in Differential Equations. Earl D. Rainville. Second edition. 1958. Macmillan Publisher, NY. CAT 58-5010
Section : Chapter 10. Nonhomogeneous Equations: Operational methods. Exercises at page 161
Problem number : 15
Date solved : Thursday, October 02, 2025 at 10:48:09 PM
CAS classification : [[_high_order, _linear, _nonhomogeneous]]

\begin{align*} y^{\left (6\right )}-y&=x^{10} \end{align*}

✓ Maple. Time used: 0.003 (sec). Leaf size: 67

ode:=diff(diff(diff(diff(diff(diff(y(x),x),x),x),x),x),x)-y(x) = x^10; 
dsolve(ode,y(x), singsol=all);
 

\[ y = \left ({\mathrm e}^{-\frac {x}{2}} c_3 +{\mathrm e}^{\frac {x}{2}} c_5 \right ) \cos \left (\frac {\sqrt {3}\, x}{2}\right )+\left ({\mathrm e}^{-\frac {x}{2}} c_4 +{\mathrm e}^{\frac {x}{2}} c_6 \right ) \sin \left (\frac {\sqrt {3}\, x}{2}\right )-x^{10}-151200 x^{4}+c_2 \,{\mathrm e}^{-x}+c_1 \,{\mathrm e}^{x} \]

✓ Mathematica. Time used: 0.003 (sec). Leaf size: 86

ode=D[y[x],{x,6}]-y[x]==x^10; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)&\to -x^{10}-151200 x^4+c_1 e^x+c_4 e^{-x}+e^{-x/2} \left (c_2 e^x+c_3\right ) \cos \left (\frac {\sqrt {3} x}{2}\right )+e^{-x/2} \left (c_6 e^x+c_5\right ) \sin \left (\frac {\sqrt {3} x}{2}\right ) \end{align*}

✓ Sympy. Time used: 0.185 (sec). Leaf size: 78

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(-x**10 - y(x) + Derivative(y(x), (x, 6)),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

\[ y{\left (x \right )} = C_{5} e^{- x} + C_{6} e^{x} - x^{10} - 151200 x^{4} + \left (C_{1} \sin {\left (\frac {\sqrt {3} x}{2} \right )} + C_{2} \cos {\left (\frac {\sqrt {3} x}{2} \right )}\right ) e^{- \frac {x}{2}} + \left (C_{3} \sin {\left (\frac {\sqrt {3} x}{2} \right )} + C_{4} \cos {\left (\frac {\sqrt {3} x}{2} \right )}\right ) e^{\frac {x}{2}} \]