4.19.40 $x^2(a^2-x^2)y^{\prime }(x{)}^2+1=0$

ODE
$$x^2(a^2-x^2)y^{\prime }(x{)}^2+1=0$$ ODE Classification

[_quadrature]

Book solution method
Missing Variables ODE, Dependent variable missing, Solve for $y^{\prime }$

Mathematica ✓
cpu = 0.0533868 (sec), leaf count = 139

$$\{\{y(x)\to c_1-\frac{ix\sqrt{x^2-a^2}\log \left(\frac{2(\sqrt{x^2-a^2}-ia)}{x}\right)}{a\sqrt{x^4-a^2{x}^2}}\},\{y(x)\to c_1+\frac{ix\sqrt{x^2-a^2}\log \left(\frac{2(\sqrt{x^2-a^2}-ia)}{x}\right)}{a\sqrt{x^4-a^2{x}^2}}\}\}$$

Maple ✓
cpu = 0.038 (sec), leaf count = 90

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

DSolve[1 + x^2*(a^2 - x^2)*y'[x]^2 == 0,y[x],x]

Mathematica raw output

{{y[x] -> C[1] - (I*x*Sqrt[-a^2 + x^2]*Log[(2*((-I)*a + Sqrt[-a^2 + x^2]))/x])/(
a*Sqrt[-(a^2*x^2) + x^4])}, {y[x] -> C[1] + (I*x*Sqrt[-a^2 + x^2]*Log[(2*((-I)*a
 + Sqrt[-a^2 + x^2]))/x])/(a*Sqrt[-(a^2*x^2) + x^4])}}

Maple raw input

dsolve(x^2*(a^2-x^2)*diff(y(x),x)^2+1 = 0, y(x),'implicit')

Maple raw output

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