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From my model I get a differential equation in which there is a term which explicitly depends on initial conditions,

the equation is of the form,

$\frac{dx(t)}{t} = a x^2 + b x(t) + f(x_0)$

Where, $x_0$ is an initial condition for the system.

I dont understand if its physible to have such an equation and if yes what does it tell me about the system. Like anything special about the system, its properties etc.

I suspect its something to do with Delay differential equations but i really have no background to understand it firmly.

Any suggestions, any hint will count a lot as i am totally blank on this one.

Thank you very much in advance.

Thanks,

Nitin

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1 Answer 1

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Once $x_0$ is fixed, you can consider $f(x_0)$ as a constant and solve the equation. If $a,b$ are constants, then the solution with initial condition $x(t_0)=x_0$ is $$ \int_{x_0}^x\frac{dz}{a\,z^2+b\,z+f(x_0)}=t-t_0. $$ In general, you could write the equation as a system: $$\begin{align} x'&=F(t,x,y)\\ y'&=0\\ x(t_0)&=x_0\\ y(t_0)&=f(x_0) \end{align}$$

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