Internal problem ID [8071]
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)]]]
Solve \begin {gather*} \boxed {y^{2} \left (y^{\prime }\right )^{2}+2 a x y y^{\prime }+\left (-a +1\right ) y^{2}+a \,x^{2}+\left (a -1\right ) b=0} \end {gather*}
✓ Solution by Maple
Time used: 0.11 (sec). Leaf size: 251
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 \relax (x ) = \sqrt {-x^{2} a +b} \\ y \relax (x ) = -\sqrt {-x^{2} a +b} \\ y \relax (x ) = \frac {\sqrt {-x^{2} a^{2}-2 a \sqrt {-b \,a^{2}+c_{1} a^{2}+b a -a c_{1}}\, x +a c_{1}+b \,a^{2}-b a}}{a} \\ y \relax (x ) = \frac {\sqrt {-x^{2} a^{2}+2 a \sqrt {-b \,a^{2}+c_{1} a^{2}+b a -a c_{1}}\, x +a c_{1}+b \,a^{2}-b a}}{a} \\ y \relax (x ) = -\frac {\sqrt {-x^{2} a^{2}-2 a \sqrt {-b \,a^{2}+c_{1} a^{2}+b a -a c_{1}}\, x +a c_{1}+b \,a^{2}-b a}}{a} \\ y \relax (x ) = -\frac {\sqrt {-x^{2} a^{2}+2 a \sqrt {-b \,a^{2}+c_{1} a^{2}+b a -a c_{1}}\, x +a c_{1}+b \,a^{2}-b a}}{a} \\ \end{align*}
✓ Solution by Mathematica
Time used: 1.024 (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*}