31.5.3 problem 3

Internal problem ID [5746]
Book : Differential Equations, By George Boole F.R.S. 1865
Section : Chapter 6
Problem number : 3
Date solved : Sunday, March 30, 2025 at 10:07:25 AM
CAS classification : [_rational, [_Riccati, _special]]

\begin{align*} u^{\prime }+u^{2}&=\frac {c}{x^{{4}/{3}}} \end{align*}

✓ Maple. Time used: 0.004 (sec). Leaf size: 39

ode:=diff(u(x),x)+u(x)^2 = c/x^(4/3); 
dsolve(ode,u(x), singsol=all);
 

\[ u = -\frac {3 c}{x^{{1}/{3}} \left (3 \tan \left (3 \sqrt {-c}\, \left (x^{{1}/{3}}-c_1 \right )\right ) x^{{1}/{3}} \sqrt {-c}+1\right )} \]

✓ Mathematica. Time used: 0.275 (sec). Leaf size: 183

ode=D[u[x],x]+u[x]^2==c*x^(-4/3); 
ic={}; 
DSolve[{ode,ic},u[x],x,IncludeSingularSolutions->True]
 

\begin{align*} u(x)\to \frac {3 c \left (3 i \sinh \left (3 \sqrt {c} \sqrt [3]{x}\right )+8 c_1 \cosh \left (3 \sqrt {c} \sqrt [3]{x}\right )\right )}{\sqrt [3]{x} \left (\left (9 i \sqrt {c} \sqrt [3]{x}-8 c_1\right ) \cosh \left (3 \sqrt {c} \sqrt [3]{x}\right )+3 \left (8 \sqrt {c} c_1 \sqrt [3]{x}-i\right ) \sinh \left (3 \sqrt {c} \sqrt [3]{x}\right )\right )} \\ u(x)\to -\frac {3 c \cosh \left (3 \sqrt {c} \sqrt [3]{x}\right )}{\sqrt [3]{x} \left (\cosh \left (3 \sqrt {c} \sqrt [3]{x}\right )-3 \sqrt {c} \sqrt [3]{x} \sinh \left (3 \sqrt {c} \sqrt [3]{x}\right )\right )} \\ \end{align*}

✗ Sympy

from sympy import * 
x = symbols("x") 
c = symbols("c") 
u = Function("u") 
ode = Eq(-c/x**(4/3) + u(x)**2 + Derivative(u(x), x),0) 
ics = {} 
dsolve(ode,func=u(x),ics=ics)
 

NotImplementedError : The given ODE -c/x**(4/3) + u(x)**2 + Derivative(u(x), x) cannot be solved by the lie group method