Internal problem ID [8826]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 1, linear first order
Problem number: 491.
ODE order: 1.
ODE degree: 2.
CAS Maple gives this as type [_rational, [_1st_order, `_with_symmetry_[F(x),G(y)]`]]
\[ \boxed {y^{2} {y^{\prime }}^{2}+2 a x y y^{\prime }+\left (-a +1\right ) y^{2}=-a \,x^{2}-\left (a -1\right ) b} \]
✓ Solution by Maple
Time used: 0.235 (sec). Leaf size: 88
dsolve(y(x)^2*diff(y(x),x)^2+2*a*x*y(x)*diff(y(x),x)+(1-a)*y(x)^2+a*x^2+(a-1)*b = 0,y(x), singsol=all)
\begin{align*} y \left (x \right ) &= \sqrt {-a \,x^{2}+b} \\ y \left (x \right ) &= -\sqrt {-a \,x^{2}+b} \\ y \left (x \right ) &= \sqrt {a \,c_{1}^{2}-2 a c_{1} x -c_{1}^{2}+2 c_{1} x -x^{2}+b} \\ y \left (x \right ) &= -\sqrt {\left (a -1\right ) c_{1}^{2}-2 x \left (a -1\right ) c_{1} -x^{2}+b} \\ \end{align*}
✓ Solution by Mathematica
Time used: 1.059 (sec). Leaf size: 65
DSolve[(-1 + a)*b + a*x^2 + (1 - a)*y[x]^2 + 2*a*x*y[x]*y'[x] + y[x]^2*y'[x]^2==0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to -\sqrt {-2 (a-1) c_1 x+(a-1) c_1{}^2+b-x^2} \\ y(x)\to \sqrt {-2 (a-1) c_1 x+(a-1) c_1{}^2+b-x^2} \\ \end{align*}