Internal problem ID [8977]
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]
\[ \boxed {y^{\prime }-\frac {\left (-y^{2}+4 a x \right )^{2}}{y}=0} \]
✓ Solution by Maple
Time used: 0.016 (sec). Leaf size: 227
dsolve(diff(y(x),x) = (-y(x)^2+4*a*x)^2/y(x),y(x), singsol=all)
\begin{align*} y \left (x \right ) &= \frac {2 \,{\mathrm e}^{4 a \,x^{2}+2 \sqrt {2}\, \sqrt {a}\, x} \sqrt {\sqrt {a}\, \left (c_{1} {\mathrm e}^{4 \sqrt {2}\, \sqrt {a}\, x}+1\right ) \left (c_{1} \left (x \sqrt {a}-\frac {\sqrt {2}}{4}\right ) {\mathrm e}^{4 \sqrt {2}\, \sqrt {a}\, x}+x \sqrt {a}+\frac {\sqrt {2}}{4}\right ) {\mathrm e}^{-8 a \,x^{2}} {\mathrm e}^{-4 \sqrt {2}\, \sqrt {a}\, x}}}{c_{1} {\mathrm e}^{4 \sqrt {2}\, \sqrt {a}\, x}+1} \\ y \left (x \right ) &= -\frac {2 \,{\mathrm e}^{4 a \,x^{2}+2 \sqrt {2}\, \sqrt {a}\, x} \sqrt {\sqrt {a}\, \left (c_{1} {\mathrm e}^{4 \sqrt {2}\, \sqrt {a}\, x}+1\right ) \left (c_{1} \left (x \sqrt {a}-\frac {\sqrt {2}}{4}\right ) {\mathrm e}^{4 \sqrt {2}\, \sqrt {a}\, x}+x \sqrt {a}+\frac {\sqrt {2}}{4}\right ) {\mathrm e}^{-8 a \,x^{2}} {\mathrm e}^{-4 \sqrt {2}\, \sqrt {a}\, x}}}{c_{1} {\mathrm e}^{4 \sqrt {2}\, \sqrt {a}\, x}+1} \\ \end{align*}
✓ Solution by Mathematica
Time used: 23.997 (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*}