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]