81.1.3 problem 1 (c)
Internal
problem
ID
[18527]
Book
:
A
short
course
on
differential
equations.
By
Donald
Francis
Campbell.
Maxmillan
company.
London.
1907
Section
:
Chapter
I.
Introduction.
Exercises
at
page
13
Problem
number
:
1
(c)
Date
solved
:
Monday, March 31, 2025 at 05:41:21 PM
CAS
classification
:
[[_3rd_order, _with_linear_symmetries]]
\begin{align*} x^{3} y^{\prime \prime \prime }+x^{2} y^{\prime \prime }+x y^{\prime }+y&=0 \end{align*}
✓ Maple. Time used: 0.003 (sec). Leaf size: 184
ode:=x^3*diff(diff(diff(y(x),x),x),x)+x^2*diff(diff(y(x),x),x)+x*diff(y(x),x)+y(x) = 0;
dsolve(ode,y(x), singsol=all);
\[
y = c_1 \,x^{-\frac {\left (188+12 \sqrt {249}\right )^{{2}/{3}}-4 \left (188+12 \sqrt {249}\right )^{{1}/{3}}-8}{6 \left (188+12 \sqrt {249}\right )^{{1}/{3}}}}+c_2 \,x^{\frac {-8+\left (188+12 \sqrt {249}\right )^{{2}/{3}}+8 \left (188+12 \sqrt {249}\right )^{{1}/{3}}}{12 \left (188+12 \sqrt {249}\right )^{{1}/{3}}}} \sin \left (\frac {\sqrt {3}\, \left (\left (188+12 \sqrt {3}\, \sqrt {83}\right )^{{2}/{3}}+8\right ) \ln \left (x \right )}{12 \left (188+12 \sqrt {3}\, \sqrt {83}\right )^{{1}/{3}}}\right )+c_3 \,x^{\frac {-8+\left (188+12 \sqrt {249}\right )^{{2}/{3}}+8 \left (188+12 \sqrt {249}\right )^{{1}/{3}}}{12 \left (188+12 \sqrt {249}\right )^{{1}/{3}}}} \cos \left (\frac {\sqrt {3}\, \left (\left (188+12 \sqrt {3}\, \sqrt {83}\right )^{{2}/{3}}+8\right ) \ln \left (x \right )}{12 \left (188+12 \sqrt {3}\, \sqrt {83}\right )^{{1}/{3}}}\right )
\]
✓ Mathematica. Time used: 0.004 (sec). Leaf size: 81
ode=x^3*D[y[x],{x,3}]+x^2*D[y[x],{x,2}]+x*D[y[x],x]+y[x]==0;
ic={};
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
\[
y(x)\to c_1 x^{\text {Root}\left [\text {$\#$1}^3-2 \text {$\#$1}^2+2 \text {$\#$1}+1\&,1\right ]}+c_3 x^{\text {Root}\left [\text {$\#$1}^3-2 \text {$\#$1}^2+2 \text {$\#$1}+1\&,3\right ]}+c_2 x^{\text {Root}\left [\text {$\#$1}^3-2 \text {$\#$1}^2+2 \text {$\#$1}+1\&,2\right ]}
\]
✓ Sympy. Time used: 0.255 (sec). Leaf size: 54
from sympy import *
x = symbols("x")
y = Function("y")
ode = Eq(x**3*Derivative(y(x), (x, 3)) + x**2*Derivative(y(x), (x, 2)) + x*Derivative(y(x), x) + y(x),0)
ics = {}
dsolve(ode,func=y(x),ics=ics)
\[
y{\left (x \right )} = \frac {C_{1}}{x^{- \operatorname {CRootOf} {\left (x^{3} - 2 x^{2} + 2 x + 1, 0\right )}}} + C_{2} x^{\operatorname {CRootOf} {\left (x^{3} - 2 x^{2} + 2 x + 1, 1\right )}} + C_{3} x^{\operatorname {CRootOf} {\left (x^{3} - 2 x^{2} + 2 x + 1, 2\right )}}
\]