4.12.50 $(x-y(x{)}^2)y^{\prime }(x)=x^2-y(x)$

ODE
$$(x-y(x{)}^2)y^{\prime }(x)=x^2-y(x)$$ ODE Classification

[_exact, _rational]

Book solution method
Exact equation

Mathematica ✓
cpu = 0.0164503 (sec), leaf count = 326

$$\{\{y(x)\to -\frac{\sqrt[3]{2}(\sqrt{(6c_1-4)x^3+9c_1^2+x^6}+3c_1+x^3){}^{2/3}+2x}{2^{2/3}\sqrt[3]{\sqrt{(6c_1-4)x^3+9c_1^2+x^6}+3c_1+x^3}}\},\{y(x)\to \frac{\sqrt[3]{2}(1-i\sqrt{3})(\sqrt{(6c_1-4)x^3+9c_1^2+x^6}+3c_1+x^3){}^{2/3}+(2+2i\sqrt{3})x}{22^{2/3}\sqrt[3]{\sqrt{(6c_1-4)x^3+9c_1^2+x^6}+3c_1+x^3}}\},\{y(x)\to \frac{\sqrt[3]{2}(1+i\sqrt{3})(\sqrt{(6c_1-4)x^3+9c_1^2+x^6}+3c_1+x^3){}^{2/3}+(2-2i\sqrt{3})x}{22^{2/3}\sqrt[3]{\sqrt{(6c_1-4)x^3+9c_1^2+x^6}+3c_1+x^3}}\}\}$$

Maple ✓
cpu = 0.013 (sec), leaf count = 20

$$\{-\frac{x^3}{3}+xy\left(x\right)-\frac{{\left(y\left(x\right)\right)}^3}{3}+\mathit{\_}\mathit{C}\mathit{1}=0\}$$ Mathematica raw input

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

Mathematica raw output

{{y[x] -> -((2*x + 2^(1/3)*(x^3 + 3*C[1] + Sqrt[x^6 + 9*C[1]^2 + x^3*(-4 + 6*C[1
])])^(2/3))/(2^(2/3)*(x^3 + 3*C[1] + Sqrt[x^6 + 9*C[1]^2 + x^3*(-4 + 6*C[1])])^(
1/3)))}, {y[x] -> ((2 + (2*I)*Sqrt[3])*x + 2^(1/3)*(1 - I*Sqrt[3])*(x^3 + 3*C[1]
 + Sqrt[x^6 + 9*C[1]^2 + x^3*(-4 + 6*C[1])])^(2/3))/(2*2^(2/3)*(x^3 + 3*C[1] + S
qrt[x^6 + 9*C[1]^2 + x^3*(-4 + 6*C[1])])^(1/3))}, {y[x] -> ((2 - (2*I)*Sqrt[3])*
x + 2^(1/3)*(1 + I*Sqrt[3])*(x^3 + 3*C[1] + Sqrt[x^6 + 9*C[1]^2 + x^3*(-4 + 6*C[
1])])^(2/3))/(2*2^(2/3)*(x^3 + 3*C[1] + Sqrt[x^6 + 9*C[1]^2 + x^3*(-4 + 6*C[1])]
)^(1/3))}}

Maple raw input

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

Maple raw output

-1/3*x^3+x*y(x)-1/3*y(x)^3+_C1 = 0