4.14.17 $(1-x^{2}y(x{)}^2)y^{\prime }(x)=y(x{)}^2(xy(x)+1)$

ODE
$$(1-x^{2}y(x{)}^2)y^{\prime }(x)=y(x{)}^2(xy(x)+1)$$ ODE Classification

[[_homogeneous, `class G`], _rational, [_Abel, `2nd type`, `class B`]]

Book solution method
Exact equation, integrating factor

Mathematica ✓
cpu = 0.0385589 (sec), leaf count = 29

$$\{\{y(x)\to -\frac{1}{x}\},\{y(x)\to -\frac{W(-e^{-c_1}x)}{x}\}\}$$

Maple ✓
cpu = 0.025 (sec), leaf count = 27

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

DSolve[(1 - x^2*y[x]^2)*y'[x] == y[x]^2*(1 + x*y[x]),y[x],x]

Mathematica raw output

{{y[x] -> -x^(-1)}, {y[x] -> -(ProductLog[-(x/E^C[1])]/x)}}

Maple raw input

dsolve((1-x^2*y(x)^2)*diff(y(x),x) = (1+x*y(x))*y(x)^2, y(x),'implicit')

Maple raw output

y(x) = -1/x, x+(-ln(y(x))-_C1)/y(x) = 0