Nonlinear differential equation gone beyond my field of expertise but I'd like to know the details of a problem and to do that I should know the general solution of the following nonlinear differential equation:

$$y'(x) = \alpha\beta e^{-\frac{x}{\gamma}} - \delta \sqrt{y(x)}$$

with $\alpha, \gamma, \delta > 0$ and $0 \leq \beta \leq 1$. Since $x$ represents the time is it also possible to assume $x \geq 0$.

I tried to solve it with Wolfram Alpha and Wolfram Mathematica but I didn't get any result due to computational time excedeed.

Is it possible to find an analytical form of $y(x)$?

  • $\begingroup$ have you tried to solve some special cases? $\endgroup$ – Dr. Sonnhard Graubner Jul 5 '16 at 15:13

I do not have a solution. But I wonder if you can gain some information about the behaviors of the equation in the extremes.

Let $y=\frac{z^2}{2}$, $$y'=zz'=\alpha\beta e^{-\frac{x}{\gamma}}-\sqrt{2}\delta z=ce^{-\frac{x}{\gamma}}-kz$$ $$zz'+kz=f(x)=ce^{-\frac{x}{\gamma}}$$

Let $w=-kx$, $$zz_w-z=-\frac{c}{k}e^{\frac{w}{\gamma k}}$$ This form is a type of Abel's equation. Unfortunately, I don't think there is a known analytic solution to this specific form. We can examine the solutions in the limits, however.

For $x\ll\gamma$, $e^{\frac{w}{\gamma k}}\approx1+\frac{w}{\gamma k}$, the text, Polyanin and Zaitsev, gives a parametric solution to

$$zz_w-z=-\frac{c}{k}(1+\frac{w}{\gamma k})$$ $$\frac{z}{t}=w+\gamma k=c_0exp(-\int\frac{t dt}{t^2-t+\frac{c}{\gamma k^2}})$$ Or $$\frac{z}{t}=-kx+\gamma k=c_0exp(-\int\frac{t dt}{t^2-t+\frac{c}{\gamma k^2}})$$

For $x\gg\gamma$, $y'=-\delta\sqrt{y}$ and $$2\sqrt{y}=-\delta x+c_1$$

There is a problem with this last equation. $\sqrt{y}>0$, but $-\delta x < 0$. If you change the requirements so that $\delta < 0$ then the solutions above hold.


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