4.19.21 $(a^2-x^2)y^{\prime }(x{)}^2+b^2=0$

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

[_quadrature]

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

Mathematica ✓
cpu = 0.0201466 (sec), leaf count = 52

$$\{\{y(x)\to c_1-b\log (\sqrt{x^2-a^2}+x)\},\{y(x)\to b\log (\sqrt{x^2-a^2}+x)+c_1\}\}$$

Maple ✓
cpu = 0.037 (sec), leaf count = 44

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

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

Mathematica raw output

{{y[x] -> C[1] - b*Log[x + Sqrt[-a^2 + x^2]]}, {y[x] -> C[1] + b*Log[x + Sqrt[-a
^2 + x^2]]}}

Maple raw input

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

Maple raw output

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