Internal problem ID [8063]
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]
\[ \boxed {a x y {y^{\prime }}^{2}-\left (a y^{2}+x^{2} b +c \right ) y^{\prime }+b x y=0} \]
✓ Solution by Maple
Time used: 0.313 (sec). Leaf size: 2733
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 \left (x \right ) = \frac {\sqrt {a \left (b \,x^{2}-c +2 x \sqrt {-b c}\right )}}{a} \\ y \left (x \right ) = \frac {\sqrt {-a \left (-b \,x^{2}+2 x \sqrt {-b c}+c \right )}}{a} \\ y \left (x \right ) = -\frac {\sqrt {a \left (b \,x^{2}-c +2 x \sqrt {-b c}\right )}}{a} \\ y \left (x \right ) = -\frac {\sqrt {-a \left (-b \,x^{2}+2 x \sqrt {-b c}+c \right )}}{a} \\ y \left (x \right ) = 0 \\ \text {Expression too large to display} \\ \text {Expression too large to display} \\ \end{align*}
✓ Solution by Mathematica
Time used: 3.621 (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*}