29.6.19 problem 165

Internal problem ID [4765]
Book : Ordinary differential equations and their solutions. By George Moseley Murphy. 1960
Section : Various 6
Problem number : 165
Date solved : Sunday, March 30, 2025 at 03:53:19 AM
CAS classification : [_rational, _Riccati]

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

✓ Maple. Time used: 0.003 (sec). Leaf size: 72

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

\[ y = \frac {\left (c_1 {| 3 x^{{1}/{3}}-1|} {\mathrm e}^{6 x^{{1}/{3}}}+\operatorname {abs}\left (1, 3 x^{{1}/{3}}-1\right ) {\mathrm e}^{6 x^{{1}/{3}}} c_1 -3 x^{{1}/{3}}\right ) x^{{1}/{3}}}{c_1 {| 3 x^{{1}/{3}}-1|} {\mathrm e}^{6 x^{{1}/{3}}}+3 x^{{1}/{3}}+1} \]

✓ Mathematica. Time used: 0.205 (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