ODE
\[ \left (a^2+x^2\right ) y'(x)^2+b-2 x y(x) y'(x)+y(x)^2=0 \] ODE Classification
[[_1st_order, _with_linear_symmetries], _rational, _Clairaut]
Book solution method
Clairaut’s equation and related types, \(f(y-x y', y')=0\)
Mathematica ✓
cpu = 0.210402 (sec), leaf count = 55
\[\left \{\left \{y(x)\to c_1 x-\sqrt {-b-a^2 c_1{}^2}\right \},\left \{y(x)\to \sqrt {-b-a^2 c_1{}^2}+c_1 x\right \}\right \}\]
Maple ✓
cpu = 0.083 (sec), leaf count = 90
\[\left [y \left (x \right ) = \frac {\sqrt {-a^{2} b -b \,x^{2}}}{a}, y \left (x \right ) = -\frac {\sqrt {-a^{2} b -b \,x^{2}}}{a}, y \left (x \right ) = x \textit {\_C1} -\sqrt {-\textit {\_C1}^{2} a^{2}-b}, y \left (x \right ) = x \textit {\_C1} +\sqrt {-\textit {\_C1}^{2} a^{2}-b}\right ]\] Mathematica raw input
DSolve[b + y[x]^2 - 2*x*y[x]*y'[x] + (a^2 + x^2)*y'[x]^2 == 0,y[x],x]
Mathematica raw output
{{y[x] -> x*C[1] - Sqrt[-b - a^2*C[1]^2]}, {y[x] -> x*C[1] + Sqrt[-b - a^2*C[1]^2]}}
Maple raw input
dsolve((a^2+x^2)*diff(y(x),x)^2-2*x*y(x)*diff(y(x),x)+b+y(x)^2 = 0, y(x))
Maple raw output
[y(x) = (-a^2*b-b*x^2)^(1/2)/a, y(x) = -(-a^2*b-b*x^2)^(1/2)/a, y(x) = x*_C1-(-_C1^2*a^2-b)^(1/2), y(x) = x*_C1+(-_C1^2*a^2-b)^(1/2)]