##### 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 = 0.263753 (sec), leaf count = 84

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

Maple
cpu = 0.357 (sec), leaf count = 135

$\left [y \left (x \right ) = -\frac {\left (\left (-2 \sqrt {\left (3 x^{\frac {1}{3}}-1\right ) \left (-1+3 \overline {x^{\frac {1}{3}}}\right )}\, {| 3 x^{\frac {1}{3}}-1|} \textit {\_C1} -3 \overline {x^{\frac {1}{3}}} \textit {\_C1} +\textit {\_C1} \right ) {\mathrm e}^{3 x^{\frac {1}{3}}}+6 \,{\mathrm e}^{-3 x^{\frac {1}{3}}} \sqrt {\left (3 x^{\frac {1}{3}}-1\right ) \left (-1+3 \overline {x^{\frac {1}{3}}}\right )}\, x^{\frac {1}{3}}\right ) x^{\frac {1}{3}}}{2 \sqrt {\left (3 x^{\frac {1}{3}}-1\right ) \left (-1+3 \overline {x^{\frac {1}{3}}}\right )}\, \left (\textit {\_C1} \,{\mathrm e}^{3 x^{\frac {1}{3}}} {| 3 x^{\frac {1}{3}}-1|}+{\mathrm e}^{-3 x^{\frac {1}{3}}} \left (3 x^{\frac {1}{3}}+1\right )\right )}\right ]$ 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))

Maple raw output

[y(x) = -1/2*((-2*((3*x^(1/3)-1)*(-1+3*conjugate(x^(1/3))))^(1/2)*abs(3*x^(1/3)-
1)*_C1-3*conjugate(x^(1/3))*_C1+_C1)*exp(3*x^(1/3))+6*exp(-3*x^(1/3))*((3*x^(1/3
)-1)*(-1+3*conjugate(x^(1/3))))^(1/2)*x^(1/3))*x^(1/3)/((3*x^(1/3)-1)*(-1+3*conj
ugate(x^(1/3))))^(1/2)/(_C1*exp(3*x^(1/3))*abs(3*x^(1/3)-1)+exp(-3*x^(1/3))*(3*x
^(1/3)+1))]