88.17.8 problem 12
Internal
problem
ID
[24119]
Book
:
Elementary
Differential
Equations.
By
Lee
Roy
Wilcox
and
Herbert
J.
Curtis.
1961
first
edition.
International
texbook
company.
Scranton,
Penn.
USA.
CAT
number
61-15976
Section
:
Chapter
5.
Special
Techniques
for
Linear
Equations.
Exercises
at
page
127
Problem
number
:
12
Date
solved
:
Thursday, October 02, 2025 at 09:59:24 PM
CAS
classification
:
[[_3rd_order, _linear, _nonhomogeneous]]
\begin{align*} y^{\prime \prime \prime }-y&=x^{n} \end{align*}
✓ Maple. Time used: 0.010 (sec). Leaf size: 134
ode:=diff(diff(diff(y(x),x),x),x)-y(x) = x^n;
dsolve(ode,y(x), singsol=all);
\[
y = -\frac {{\mathrm e}^{-\frac {x}{2}} \left (\sqrt {3}\, \int {\mathrm e}^{\frac {x}{2}} \left (\sqrt {3}\, \cos \left (\frac {\sqrt {3}\, x}{2}\right )-3 \sin \left (\frac {\sqrt {3}\, x}{2}\right )\right ) x^{n}d x \cos \left (\frac {\sqrt {3}\, x}{2}\right )+\sqrt {3}\, \int {\mathrm e}^{\frac {x}{2}} \left (\sqrt {3}\, \sin \left (\frac {\sqrt {3}\, x}{2}\right )+3 \cos \left (\frac {\sqrt {3}\, x}{2}\right )\right ) x^{n}d x \sin \left (\frac {\sqrt {3}\, x}{2}\right )-9 c_2 \cos \left (\frac {\sqrt {3}\, x}{2}\right )-9 c_3 \sin \left (\frac {\sqrt {3}\, x}{2}\right )\right )}{9}+\frac {\left (\int x^{n} {\mathrm e}^{-x}d x +3 c_1 \right ) {\mathrm e}^{x}}{3}
\]
✓ Mathematica. Time used: 0.815 (sec). Leaf size: 228
ode=D[y[x],{x,3}]-y[x]==x^n;
ic={};
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
\begin{align*} y(x)&\to \frac {1}{3} e^{-x/2} \left (-2^n 3^{n/2} x^n \left (-\left (\left (\sqrt {3}-3 i\right ) x\right )\right )^{-n} \Gamma \left (n+1,\frac {1}{2} i \left (i+\sqrt {3}\right ) x\right ) \left (\cos \left (\frac {\sqrt {3} x}{2}\right )+i \sin \left (\frac {\sqrt {3} x}{2}\right )\right )-2^n 3^{n/2} x^n \left (-\left (\left (\sqrt {3}+3 i\right ) x\right )\right )^{-n} \Gamma \left (n+1,-\frac {1}{2} i \left (-i+\sqrt {3}\right ) x\right ) \left (\cos \left (\frac {\sqrt {3} x}{2}\right )-i \sin \left (\frac {\sqrt {3} x}{2}\right )\right )-e^{3 x/2} \Gamma (n+1,x)+3 \left (c_1 e^{3 x/2}+c_2 \cos \left (\frac {\sqrt {3} x}{2}\right )+c_3 \sin \left (\frac {\sqrt {3} x}{2}\right )\right )\right ) \end{align*}
✓ Sympy. Time used: 16.920 (sec). Leaf size: 129
from sympy import *
x = symbols("x")
n = symbols("n")
y = Function("y")
ode = Eq(-x**n - y(x) + Derivative(y(x), (x, 3)),0)
ics = {}
dsolve(ode,func=y(x),ics=ics)
\[
y{\left (x \right )} = \left (\left (C_{1} - \frac {2 \int x^{n} e^{\frac {x}{2}} \sin {\left (\frac {3 \sqrt {3} x + 2 \pi }{6} \right )}\, dx}{3}\right ) \sin {\left (\frac {\sqrt {3} x}{2} \right )} + \left (C_{2} - \frac {2 \int x^{n} e^{\frac {x}{2}} \cos {\left (\frac {3 \sqrt {3} x + 2 \pi }{6} \right )}\, dx}{3}\right ) \cos {\left (\frac {\sqrt {3} x}{2} \right )}\right ) e^{- \frac {x}{2}} + \left (C_{3} + \frac {n \Gamma \left (n + 1\right ) \gamma \left (n + 1, x\right )}{3 \Gamma \left (n + 2\right )} + \frac {\Gamma \left (n + 1\right ) \gamma \left (n + 1, x\right )}{3 \Gamma \left (n + 2\right )}\right ) e^{x}
\]