56.1.37 problem 38

Internal problem ID [8749]
Book : Own collection of miscellaneous problems
Section : section 1.0
Problem number : 38
Date solved : Wednesday, March 05, 2025 at 06:43:59 AM
CAS classification : [_rational, _Riccati]

\begin{align*} x y^{\prime }-y+y^{2}&=x^{{2}/{3}} \end{align*}

✓ Maple. Time used: 0.001 (sec). Leaf size: 72

ode:=x*diff(y(x),x)-y(x)+y(x)^2 = x^(2/3); 
dsolve(ode,y(x), singsol=all);
 

\[ y = \frac {x^{{1}/{3}} \left (c_{1} {\mathrm e}^{6 x^{{1}/{3}}} \operatorname {abs}\left (1, 3 x^{{1}/{3}}-1\right )+c_{1} {\mathrm e}^{6 x^{{1}/{3}}} {| 3 x^{{1}/{3}}-1|}-3 x^{{1}/{3}}\right )}{c_{1} {\mathrm e}^{6 x^{{1}/{3}}} {| 3 x^{{1}/{3}}-1|}+3 x^{{1}/{3}}+1} \]

✓ Mathematica. Time used: 0.2 (sec). Leaf size: 131

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

\begin{align*} y(x)\to \frac {3 x^{2/3} \left (c_1 \cosh \left (3 \sqrt [3]{x}\right )-i \sinh \left (3 \sqrt [3]{x}\right )\right )}{\left (-3 i \sqrt [3]{x}-c_1\right ) \cosh \left (3 \sqrt [3]{x}\right )+\left (3 c_1 \sqrt [3]{x}+i\right ) \sinh \left (3 \sqrt [3]{x}\right )} \\ y(x)\to \frac {3 x^{2/3} \cosh \left (3 \sqrt [3]{x}\right )}{3 \sqrt [3]{x} \sinh \left (3 \sqrt [3]{x}\right )-\cosh \left (3 \sqrt [3]{x}\right )} \\ \end{align*}

✗ Sympy

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

NotImplementedError : The given ODE Derivative(y(x), x) + y(x)**2/x - y(x)/x - 1/x**(1/3) cannot be solved by the factorable group method