ODE
\[ (a-b) y(x)^2 y'(x)^2-a b+a y(x)^2-b x^2-2 b x y(x) y'(x)=0 \] ODE Classification
[_rational, [_1st_order, `_with_symmetry_[F(x),G(y)]`]]
Book solution method
Change of variable
Mathematica ✓
cpu = 1.1851 (sec), leaf count = 86
\[\left \{\left \{y(x)\to -\frac {\sqrt {b \left (b-x^2\right )+a \left (-b+(x-c_1){}^2\right )}}{\sqrt {b-a}}\right \},\left \{y(x)\to \frac {\sqrt {b \left (b-x^2\right )+a \left (-b+(x-c_1){}^2\right )}}{\sqrt {b-a}}\right \}\right \}\]
Maple ✓
cpu = 2.43 (sec), leaf count = 260
\[\left [y \left (x \right ) = \frac {\sqrt {\left (a -b \right ) b \left (x^{2}+a -b \right )}}{a -b}, y \left (x \right ) = -\frac {\sqrt {\left (a -b \right ) b \left (x^{2}+a -b \right )}}{a -b}, y \left (x \right ) = \frac {\sqrt {-\textit {\_C1} a b +\textit {\_C1} \,b^{2}-b^{2} x^{2}-2 b \sqrt {\textit {\_C1} a b -a \,b^{2}}\, x +a \,b^{2}}}{b}, y \left (x \right ) = \frac {\sqrt {-\textit {\_C1} a b +\textit {\_C1} \,b^{2}-b^{2} x^{2}+2 b \sqrt {\textit {\_C1} a b -a \,b^{2}}\, x +a \,b^{2}}}{b}, y \left (x \right ) = -\frac {\sqrt {-\textit {\_C1} a b +\textit {\_C1} \,b^{2}-b^{2} x^{2}-2 b \sqrt {\textit {\_C1} a b -a \,b^{2}}\, x +a \,b^{2}}}{b}, y \left (x \right ) = -\frac {\sqrt {-\textit {\_C1} a b +\textit {\_C1} \,b^{2}-b^{2} x^{2}+2 b \sqrt {\textit {\_C1} a b -a \,b^{2}}\, x +a \,b^{2}}}{b}\right ]\] Mathematica raw input
DSolve[-(a*b) - b*x^2 + a*y[x]^2 - 2*b*x*y[x]*y'[x] + (a - b)*y[x]^2*y'[x]^2 == 0,y[x],x]
Mathematica raw output
{{y[x] -> -(Sqrt[b*(b - x^2) + a*(-b + (x - C[1])^2)]/Sqrt[-a + b])}, {y[x] -> Sqrt[b*(b - x^2) + a*(-b + (x - C[1])^2)]/Sqrt[-a + b]}}
Maple raw input
dsolve((a-b)*y(x)^2*diff(y(x),x)^2-2*b*x*y(x)*diff(y(x),x)-a*b-b*x^2+a*y(x)^2 = 0, y(x))
Maple raw output
[y(x) = 1/(a-b)*((a-b)*b*(x^2+a-b))^(1/2), y(x) = -1/(a-b)*((a-b)*b*(x^2+a-b))^(1/2), y(x) = 1/b*(-_C1*a*b+_C1*b^2-b^2*x^2-2*b*(_C1*a*b-a*b^2)^(1/2)*x+a*b^2)^(1/2), y(x) = 1/b*(-_C1*a*b+_C1*b^2-b^2*x^2+2*b*(_C1*a*b-a*b^2)^(1/2)*x+a*b^2)^(1/2), y(x) = -1/b*(-_C1*a*b+_C1*b^2-b^2*x^2-2*b*(_C1*a*b-a*b^2)^(1/2)*x+a*b^2)^(1/2), y(x) = -1/b*(-_C1*a*b+_C1*b^2-b^2*x^2+2*b*(_C1*a*b-a*b^2)^(1/2)*x+a*b^2)^(1/2)]