ODE
\[ \left (a^2-x^2\right ) y'(x)^2+b^2=0 \] ODE Classification
[_quadrature]
Book solution method
Missing Variables ODE, Dependent variable missing, Solve for \(y'\)
Mathematica ✓
cpu = 0.161804 (sec), leaf count = 109
\[\left \{\left \{y(x)\to \frac {1}{2} b \log \left (1-\frac {x}{\sqrt {x^2-a^2}}\right )-\frac {1}{2} b \log \left (\frac {x}{\sqrt {x^2-a^2}}+1\right )+c_1\right \},\left \{y(x)\to -\frac {1}{2} b \log \left (1-\frac {x}{\sqrt {x^2-a^2}}\right )+\frac {1}{2} b \log \left (\frac {x}{\sqrt {x^2-a^2}}+1\right )+c_1\right \}\right \}\]
Maple ✓
cpu = 0.112 (sec), leaf count = 44
\[\left [y \left (x \right ) = b \ln \left (x +\sqrt {-a^{2}+x^{2}}\right )+\textit {\_C1}, y \left (x \right ) = -b \ln \left (x +\sqrt {-a^{2}+x^{2}}\right )+\textit {\_C1}\right ]\] Mathematica raw input
DSolve[b^2 + (a^2 - x^2)*y'[x]^2 == 0,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 = 0, 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]