4.4.18 \(x y'(x)+y(x)^2-y(x)=x^{2/3}\)

ODE
\[ x y'(x)+y(x)^2-y(x)=x^{2/3} \] ODE Classification

[_rational, _Riccati]

Book solution method
Riccati ODE, Special cases

Mathematica
cpu = 0.01117 (sec), leaf count = 84

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

Maple
cpu = 0.35 (sec), leaf count = 86

\[ \left \{ y \relax (x ) ={1 \left ({\it \_C1}\,{{\rm e}^{3\,\sqrt [3]{x}}} \left | 3\,\sqrt [3]{x}-1 \right | +{\it \_C1}\,{{\rm e}^{3\,\sqrt [3]{x}}} \left | 1 \right | -3\,{{\rm e}^{-3\,\sqrt [3]{x}}}\sqrt [3]{x} \right ) \sqrt [3]{x} \left ({\it \_C1}\,{{\rm e}^{3\,\sqrt [3]{x}}} \left | 3\,\sqrt [3]{x}-1 \right | +{{\rm e}^{-3\,\sqrt [3]{x}}} \left (3\,\sqrt [3]{x}+1 \right ) \right ) ^{-1}} \right \} \] Mathematica raw input

DSolve[-y[x] + y[x]^2 + x*y'[x] == x^(2/3),y[x],x]

Mathematica raw output

{{y[x] -> (3*x^(2/3)*(C[1]*Cosh[3*x^(1/3)] - I*Sinh[3*x^(1/3)]))/(((-3*I)*x^(1/3
) - C[1])*Cosh[3*x^(1/3)] + (I + 3*x^(1/3)*C[1])*Sinh[3*x^(1/3)])}}

Maple raw input

dsolve(x*diff(y(x),x)-y(x)+y(x)^2 = x^(2/3), y(x),'implicit')

Maple raw output

y(x) = (_C1*exp(3*x^(1/3))*abs(3*x^(1/3)-1)+_C1*exp(3*x^(1/3))*abs(1,3*x^(1/3)-1
)-3*exp(-3*x^(1/3))*x^(1/3))*x^(1/3)/(_C1*exp(3*x^(1/3))*abs(3*x^(1/3)-1)+exp(-3
*x^(1/3))*(3*x^(1/3)+1))