4.19.20 $(a^2+x^2)y^{\prime }(x{)}^2=b^2$

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

[_quadrature]

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

Mathematica ✓
cpu = 0.0179269 (sec), leaf count = 48

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

Maple ✓
cpu = 0.032 (sec), leaf count = 40

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

DSolve[(a^2 + x^2)*y'[x]^2 == b^2,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, 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