31.5.5 problem 5
Internal
problem
ID
[5748]
Book
:
Differential
Equations,
By
George
Boole
F.R.S.
1865
Section
:
Chapter
6
Problem
number
:
5
Date
solved
:
Sunday, March 30, 2025 at 10:07:28 AM
CAS
classification
:
[_rational, [_Riccati, _special]]
\begin{align*} u^{\prime }-u^{2}&=\frac {2}{x^{{8}/{3}}} \end{align*}
✓ Maple. Time used: 0.007 (sec). Leaf size: 78
ode:=diff(u(x),x)-u(x)^2 = 2/x^(8/3);
dsolve(ode,u(x), singsol=all);
\[
u = -\frac {3 \left (\tan \left (3 \sqrt {2}\, \left (\left (\frac {1}{x}\right )^{{1}/{3}}-c_1 \right )\right ) \sqrt {2}\, x \left (\frac {1}{x}\right )^{{2}/{3}}+\frac {x \left (\frac {1}{x}\right )^{{1}/{3}}}{3}-2\right )}{\left (\frac {1}{x}\right )^{{1}/{3}} x^{2} \left (3 \left (\frac {1}{x}\right )^{{1}/{3}} \sqrt {2}\, \tan \left (3 \sqrt {2}\, \left (\left (\frac {1}{x}\right )^{{1}/{3}}-c_1 \right )\right )+1\right )}
\]
✓ Mathematica. Time used: 0.244 (sec). Leaf size: 215
ode=D[u[x],x]-u[x]^2==x^(-8/3);
ic={};
DSolve[{ode,ic},u[x],x,IncludeSingularSolutions->True]
\begin{align*}
u(x)\to -\frac {\left (-9 \sqrt [3]{\frac {1}{x}}+c_1 \left (8-24 \left (\frac {1}{x}\right )^{2/3}\right )\right ) \cos \left (3 \sqrt [3]{\frac {1}{x}}\right )+3 \left (-3 \left (\frac {1}{x}\right )^{2/3}+8 c_1 \sqrt [3]{\frac {1}{x}}+1\right ) \sin \left (3 \sqrt [3]{\frac {1}{x}}\right )}{x \left (\left (-9 \sqrt [3]{\frac {1}{x}}+8 c_1\right ) \cos \left (3 \sqrt [3]{\frac {1}{x}}\right )+3 \left (1+8 c_1 \sqrt [3]{\frac {1}{x}}\right ) \sin \left (3 \sqrt [3]{\frac {1}{x}}\right )\right )} \\
u(x)\to \frac {\left (3 \left (\frac {1}{x}\right )^{2/3}-1\right ) \cos \left (3 \sqrt [3]{\frac {1}{x}}\right )-3 \sqrt [3]{\frac {1}{x}} \sin \left (3 \sqrt [3]{\frac {1}{x}}\right )}{x \left (3 \sqrt [3]{\frac {1}{x}} \sin \left (3 \sqrt [3]{\frac {1}{x}}\right )+\cos \left (3 \sqrt [3]{\frac {1}{x}}\right )\right )} \\
\end{align*}
✗ Sympy
from sympy import *
x = symbols("x")
u = Function("u")
ode = Eq(-u(x)**2 + Derivative(u(x), x) - 2/x**(8/3),0)
ics = {}
dsolve(ode,func=u(x),ics=ics)
NotImplementedError : The given ODE -u(x)**2 + Derivative(u(x), x) - 2/x**(8/3) cannot be solved by the lie group method