Internal problem ID [8065]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 1, linear first order
Problem number: 485.
ODE order: 1.
ODE degree: 2.
CAS Maple gives this as type [_rational]
Solve \begin {gather*} \boxed {a x y \left (y^{\prime }\right )^{2}-\left (a y^{2}+b \,x^{2}+c \right ) y^{\prime }+b x y=0} \end {gather*}
✓ Solution by Maple
Time used: 0.483 (sec). Leaf size: 2664
dsolve(a*x*y(x)*diff(y(x),x)^2-(a*y(x)^2+b*x^2+c)*diff(y(x),x)+b*x*y(x) = 0,y(x), singsol=all)
\begin{align*} y \relax (x ) = 0 \\ \text {Expression too large to display} \\ \text {Expression too large to display} \\ y \relax (x ) = \frac {\sqrt {a \left (a c_{1} x +b \,x^{2}-c \right )}}{a} \\ y \relax (x ) = -\frac {\sqrt {a \left (a c_{1} x +b \,x^{2}-c \right )}}{a} \\ \end{align*}
✓ Solution by Mathematica
Time used: 0.359 (sec). Leaf size: 155
DSolve[b*x*y[x] - (c + b*x^2 + a*y[x]^2)*y'[x] + a*x*y[x]*y'[x]^2==0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \sqrt {c_1 \left (x^2+\frac {c}{b-a c_1}\right )} \\ y(x)\to -\sqrt {-\frac {\left (\sqrt {c}+i \sqrt {b} x\right )^2}{a}} \\ y(x)\to \sqrt {-\frac {\left (\sqrt {c}+i \sqrt {b} x\right )^2}{a}} \\ y(x)\to -\sqrt {\frac {\left (\sqrt {b} x+i \sqrt {c}\right )^2}{a}} \\ y(x)\to \sqrt {\frac {\left (\sqrt {b} x+i \sqrt {c}\right )^2}{a}} \\ \end{align*}