25.5.5 problem 5

Internal problem ID [6883]
Book : Differential Equations, By George Boole F.R.S. 1865
Section : Chapter 6
Problem number : 5
Date solved : Tuesday, September 30, 2025 at 03:59:56 PM
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: 0.155 (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