Internal problem ID [8080]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 1, linear first order
Problem number: 500.
ODE order: 1.
ODE degree: 2.
CAS Maple gives this as type [_rational, [_1st_order, _with_symmetry_[F(x),G(y)]]]
Solve \begin {gather*} \boxed {\left (a -b \right ) y^{2} \left (y^{\prime }\right )^{2}-2 b x y y^{\prime }+a y^{2}-x^{2} b -a b=0} \end {gather*}
✓ Solution by Maple
Time used: 0.062 (sec). Leaf size: 260
dsolve((a-b)*y(x)^2*diff(y(x),x)^2-2*b*x*y(x)*diff(y(x),x)+a*y(x)^2-b*x^2-a*b = 0,y(x), singsol=all)
\begin{align*} y \relax (x ) = \frac {\sqrt {\left (a -b \right ) b \left (x^{2}+a -b \right )}}{a -b} \\ y \relax (x ) = -\frac {\sqrt {\left (a -b \right ) b \left (x^{2}+a -b \right )}}{a -b} \\ y \relax (x ) = \frac {\sqrt {-c_{1} b a +c_{1} b^{2}-b^{2} x^{2}-2 b \sqrt {-a \,b^{2}+c_{1} b a}\, x +a \,b^{2}}}{b} \\ y \relax (x ) = \frac {\sqrt {-c_{1} b a +c_{1} b^{2}-b^{2} x^{2}+2 b \sqrt {-a \,b^{2}+c_{1} b a}\, x +a \,b^{2}}}{b} \\ y \relax (x ) = -\frac {\sqrt {-c_{1} b a +c_{1} b^{2}-b^{2} x^{2}-2 b \sqrt {-a \,b^{2}+c_{1} b a}\, x +a \,b^{2}}}{b} \\ y \relax (x ) = -\frac {\sqrt {-c_{1} b a +c_{1} b^{2}-b^{2} x^{2}+2 b \sqrt {-a \,b^{2}+c_{1} b a}\, x +a \,b^{2}}}{b} \\ \end{align*}
✓ Solution by Mathematica
Time used: 1.341 (sec). Leaf size: 86
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,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to -\frac {\sqrt {b \left (b-x^2\right )+a \left (-b+(x-c_1){}^2\right )}}{\sqrt {b-a}} \\ y(x)\to \frac {\sqrt {b \left (b-x^2\right )+a \left (-b+(x-c_1){}^2\right )}}{\sqrt {b-a}} \\ \end{align*}