4.12.12 $axy(x)y^{\prime }(x)+x^2-y(x{)}^2=0$

ODE
$$axy(x)y^{\prime }(x)+x^2-y(x{)}^2=0$$ ODE Classification

[[_homogeneous, `class A`], _rational, _Bernoulli]

Book solution method
Homogeneous equation

Mathematica ✓
cpu = 0.011585 (sec), leaf count = 72

$$\{\{y(x)\to -\frac{\sqrt{(a-1)c_1{x}^{2/a}-x^2}}{\sqrt{a-1}}\},\{y(x)\to \frac{\sqrt{(a-1)c_1{x}^{2/a}-x^2}}{\sqrt{a-1}}\}\}$$

Maple ✓
cpu = 0.008 (sec), leaf count = 27

$$\{\frac{x^2}{a-1}-x^{2a^{-1}}\mathit{\_}\mathit{C}\mathit{1}+{\left(y\left(x\right)\right)}^2=0\}$$ Mathematica raw input

DSolve[x^2 - y[x]^2 + a*x*y[x]*y'[x] == 0,y[x],x]

Mathematica raw output

{{y[x] -> -(Sqrt[-x^2 + (-1 + a)*x^(2/a)*C[1]]/Sqrt[-1 + a])}, {y[x] -> Sqrt[-x^
2 + (-1 + a)*x^(2/a)*C[1]]/Sqrt[-1 + a]}}

Maple raw input

dsolve(a*x*y(x)*diff(y(x),x)+x^2-y(x)^2 = 0, y(x),'implicit')

Maple raw output

x^2/(a-1)-x^(2/a)*_C1+y(x)^2 = 0