##### 4.19.20 $$\left (a^2+x^2\right ) y'(x)^2=b^2$$

ODE
$\left (a^2+x^2\right ) y'(x)^2=b^2$ ODE Classiﬁcation

[_quadrature]

Book solution method
Missing Variables ODE, Dependent variable missing, Solve for $$y'$$

Mathematica
cpu = 0.164934 (sec), leaf count = 101

$\left \{\left \{y(x)\to \frac {1}{2} b \log \left (1-\frac {x}{\sqrt {a^2+x^2}}\right )-\frac {1}{2} b \log \left (\frac {x}{\sqrt {a^2+x^2}}+1\right )+c_1\right \},\left \{y(x)\to -\frac {1}{2} b \log \left (1-\frac {x}{\sqrt {a^2+x^2}}\right )+\frac {1}{2} b \log \left (\frac {x}{\sqrt {a^2+x^2}}+1\right )+c_1\right \}\right \}$

Maple
cpu = 0.071 (sec), leaf count = 40

$\left [y \left (x \right ) = b \ln \left (x +\sqrt {a^{2}+x^{2}}\right )+\textit {\_C1}, y \left (x \right ) = -b \ln \left (x +\sqrt {a^{2}+x^{2}}\right )+\textit {\_C1}\right ]$ 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[1 - x/Sqrt[a^2 + x^2]])/2 - (b*Log[1 + x/Sqrt[a^2 + x^2]
])/2}, {y[x] -> C[1] - (b*Log[1 - x/Sqrt[a^2 + x^2]])/2 + (b*Log[1 + x/Sqrt[a^2
+ x^2]])/2}}

Maple raw input

dsolve((a^2+x^2)*diff(y(x),x)^2 = b^2, y(x))

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]