Internal problem ID [8222]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 1, Additional non-linear first order
Problem number: 642.
ODE order: 1.
ODE degree: 1.
CAS Maple gives this as type [_rational]
Solve \begin {gather*} \boxed {y^{\prime }-\frac {\left (-y^{2}+4 a x \right )^{2}}{y}=0} \end {gather*}
✓ Solution by Maple
Time used: 0.062 (sec). Leaf size: 286
dsolve(diff(y(x),x) = (-y(x)^2+4*a*x)^2/y(x),y(x), singsol=all)
\begin{align*} y \relax (x ) = \frac {\sqrt {4}\, \sqrt {\left (c_{1} {\mathrm e}^{2 x \left (\sqrt {2}\, \sqrt {a}-2 x a \right )}+{\mathrm e}^{-2 x \left (\sqrt {2}\, \sqrt {a}+2 x a \right )}\right ) \left (c_{1} \left (x a -\frac {\sqrt {2}\, \sqrt {a}}{4}\right ) {\mathrm e}^{2 x \left (\sqrt {2}\, \sqrt {a}-2 x a \right )}+{\mathrm e}^{-2 x \left (\sqrt {2}\, \sqrt {a}+2 x a \right )} \left (x a +\frac {\sqrt {2}\, \sqrt {a}}{4}\right )\right )}}{c_{1} {\mathrm e}^{2 x \left (\sqrt {2}\, \sqrt {a}-2 x a \right )}+{\mathrm e}^{-2 x \left (\sqrt {2}\, \sqrt {a}+2 x a \right )}} \\ y \relax (x ) = -\frac {\sqrt {4}\, \sqrt {\left (c_{1} {\mathrm e}^{2 x \left (\sqrt {2}\, \sqrt {a}-2 x a \right )}+{\mathrm e}^{-2 x \left (\sqrt {2}\, \sqrt {a}+2 x a \right )}\right ) \left (c_{1} \left (x a -\frac {\sqrt {2}\, \sqrt {a}}{4}\right ) {\mathrm e}^{2 x \left (\sqrt {2}\, \sqrt {a}-2 x a \right )}+{\mathrm e}^{-2 x \left (\sqrt {2}\, \sqrt {a}+2 x a \right )} \left (x a +\frac {\sqrt {2}\, \sqrt {a}}{4}\right )\right )}}{c_{1} {\mathrm e}^{2 x \left (\sqrt {2}\, \sqrt {a}-2 x a \right )}+{\mathrm e}^{-2 x \left (\sqrt {2}\, \sqrt {a}+2 x a \right )}} \\ \end{align*}
✓ Solution by Mathematica
Time used: 14.258 (sec). Leaf size: 95
DSolve[y'[x] == (4*a*x - y[x]^2)^2/y[x],y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to -\sqrt {4 a x-\sqrt {2} \sqrt {a} \tanh \left (\frac {\sqrt {2} (2 a x-c_1)}{\sqrt {a}}\right )} \\ y(x)\to \sqrt {4 a x-\sqrt {2} \sqrt {a} \tanh \left (\frac {\sqrt {2} (2 a x-c_1)}{\sqrt {a}}\right )} \\ \end{align*}