4.15.37 $(x-y(x{)}^2\sqrt{y(x{)}^2-x^2})y^{\prime }(x)=y(x)(x\sqrt{y(x{)}^2-x^2}+1)$

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

[NONE]

Book solution method
Exact equation, integrating factor

Mathematica ✓
cpu = 3.75411 (sec), leaf count = 35

$$\text{Solve}[2{\tan }^{-1}\left(\frac{x}{\sqrt{y(x{)}^2-x^2}}\right)+x^2+y(x{)}^2=2c_1,y(x)]$$

Maple ✓
cpu = 0.413 (sec), leaf count = 73

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

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

Mathematica raw output

Solve[x^2 + 2*ArcTan[x/Sqrt[-x^2 + y[x]^2]] + y[x]^2 == 2*C[1], y[x]]

Maple raw input

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

Maple raw output

1/2*(2*x*ln((-2*x^2+2*(-x^2)^(1/2)*(y(x)^2-x^2)^(1/2))/y(x))+(x^2+y(x)^2-2*_C1)*
(-x^2)^(1/2)-2*x*ln(x))/(-x^2)^(1/2) = 0