4.12.33 $2x(x+1)y(x)y^{\prime }(x)=y(x{)}^2+1$

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

[_separable]

Book solution method
Separable ODE, Neither variable missing

Mathematica ✓
cpu = 0.0174176 (sec), leaf count = 58

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

Maple ✓
cpu = 0.011 (sec), leaf count = 23

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

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

Mathematica raw output

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

Maple raw input

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

Maple raw output

((1+x)*y(x)^2-_C1*x+1)/(1+x) = 0