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I came across this first order differential equation $$ f'(x) = \left( \frac{1}{x} + \frac{g'(x)}{g(x)} \right) f(x) - c \frac{g'(x)}{g(x)} \textrm{,}$$ where $g(x)$ is this logistic type function $$ g(x) = \frac{h}{1+e^{-(x-a)}} + b $$ and $a$, $b$, $c$ and $h$ are constants.

1st try:

First I tried integrating both sides and got $$ f(x) = \int \frac{f(x)}{x} \mathrm{d}x + \left[ cx - f(x) \right] \ln g(x) + C $$ or $$ \int \frac{f(x)}{x} \mathrm{d}x = \left[ 1 + \ln g(x) \right] f(x) - c x \ln g(x) + C $$ and in both cases the leftover integral term seems to be quite problematic.

2nd try:

After that I remembered that I should probably be using the integrating factor method and looked up that the solution should be $$ f(x) = \frac{1}{\mu(x)} \left( \int \mu(x) \left( - c \frac{g'(x)}{g(x)}\right) \mathrm{d} x + C\right) $$ with $$ \mu(x) = \exp \left[ -\int \left( \frac{g'(x)}{g(x)} + \frac{1}{x} \right) \mathrm{d}x \right] = \frac{1}{x g(x)}$$ so finally $$ f(x) = x g(x) \left( \int \frac{1}{x g(x)} \left( - c \frac{g'(x)}{g(x)}\right) \mathrm{d} x + C \right) = - c x g(x) \left( \int \frac{g'(x)}{g^2(x)} \frac{\mathrm{d} x}{x} + C\right) \textrm{.}$$

The evaluation of the integral $$ \int \frac{g'(x)}{g^2(x)} \frac{\mathrm{d} x}{x} = \int \frac{\mathrm{d}}{\mathrm{d}x} \left( \frac{h}{1+e^{-(x-a)}} + b \right) \left( \frac{h}{1+e^{-(x-a)}} + b \right)^{-2} \frac{\mathrm{d} x}{x} $$ $$ = h \int \left( \frac{ e^{-(x-a)}}{(1+e^{-(x-a)})^2} \right) \left( \frac{h}{1+e^{-(x-a)}} + b \right)^{-2} \frac{\mathrm{d} x}{x} $$ seems quite though and I haven't made much progress.

Is there a way to evaluate this integral or is there possibly an easier way to solve the original differential equation?

EDIT: I just realized that $$ g'(x) = [g(x) - b] - [g(x) - b]^2 = g(x) - b - g^2(x) + 2g(x)b - b^2 $$ so that $$ \frac{g'(x)}{g^2(x)} = \frac{1 + 2b}{g(x)} - \frac{b + b^2}{g^2(x)} - 1 $$ and I would only need to know how to evaluate $$ \int \frac{1}{g(x)} \frac{\mathrm{d} x}{x} \quad \textrm{and} \quad \int \frac{1}{g^2(x)} \frac{\mathrm{d} x}{x} \textrm{.} $$ Could there be a closed form solution or maybe just a series representation or something?

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Whenever I see $\displaystyle\frac{g'(x)}{g(x)}$, my immediate instinct is to write that as $[\ln g(x)]'$. Here it helps, as follows:

$f'(x) = [\,\displaystyle\frac{1}{x} + (\ln g)'\,]\,f(x) - c\,(\ln g)'$

$f' = \displaystyle\frac{f}{x} + (f-c)\,(\ln g)'$

Now define a new function $u(x) = f(x) - c$. Then,

$u' = \displaystyle\frac{u + c}{x} + (\ln g)'\,u$

$x\,u' = u + c + x\,(\ln g)'\,u = c + \underbrace{[1 + x\,(\ln g)']}_{a(x)}\,u$

$x\,u'(x) - a(x)\,u(x) = c$

The homogeneous equation is:

$\displaystyle\frac{u'(x)}{u(x)} = \displaystyle\frac{a(x)}{x}$

So...

$[\,\ln u(x)\,]' = \displaystyle\frac{a(x)}{x} = \displaystyle\frac{1}{x} + [\,\ln g(x)\,]'$

and

$\ln u = \ln x + \ln g + \ln A$

($A$ is an integration constant)

$u(x) = A\,x\,g(x)$

I'll let you continue from here. You could use the method of variation of parameters to find a particular solution $u_{p}(x)$ of

$x\,u'(x) - a(x)\,u(x) = c$

then use $f(x) = A\,x\,g(x) + u_{p}(x) + c$ as your final general solution.

Edit: I decided to complete the solution so...

Method of variation of parameters. We have a general solution of the homogeneous equation, $u(x) = A\,x\,g(x)$. Now, try a particular solution of the form $u_p(x) = A(x)\,x\,g(x)$. Then,

$x\,u'(x) - a(x)\,u(x) = c$

gives us

$A'(x)\,x^2\,g(x) = c$

so

$A(x) = \displaystyle\int^x \frac{c\,dt}{t^2\,g(t)}$

and

$u_p(x) = A(x)\,x\,g(x) = c\,x\,g(x) \displaystyle\int^x \frac{dt}{t^2\,g(t)}$

and the final general solution is

$f(x) = A\,x\,g(x) + c\,x\,g(x) \displaystyle\int^x \frac{dt}{t^2\,g(t)} + c$

where $A$ is a constant.

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  • $\begingroup$ Thanks for the answer, any ideas on the evaluation of $$ \int^x \frac{\mathrm{d} t}{t^2 g(t)} = \int^x \frac{1}{h/(1+\exp[-(t-a)]) + b} \frac{\mathrm{d} t}{t^2} \textrm{?} $$ $\endgroup$ – 655321 Apr 23 '15 at 22:35
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    $\begingroup$ Yes. The integral simplifies into the sum of two integrals of the form $\int^x dt\,\displaystyle\frac{t^{-2}\,(e^{t-a})^m}{B\,e^{t-a} + 1}$, one with $m=0$ and the other with $m=1$, where $B=1+h/b$. Then try to use integration by parts to solve the two integrals. $\endgroup$ – wltrup Apr 24 '15 at 4:45

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