First order nonlinear differential equation

[heatherw]

Hi, I've come across this equation in my research, and I am so far unable to solve it.

The equation is:
$$\frac{dV}{dt} + \frac{1}{RC}V - \frac{I}{C}e^{(-V/n)} = \frac{V_f}{RC} - \frac{I}{C}$$

both terms on the right-hand-side are constants; throughout the equation R, C, I, n, and V_f are constants. I want to solve for V. The equation is derived from a pretty simple resistor, capacitor, diode circuit, which I can describe if anyone thinks that will help.

I can solve the equation without the $$\frac{I}{C}e^{(-V/n)}$$ but I just have no idea how to do it with the exponential term.

Thanks for any help.

 

[lippyka]

This type of equation is known as a differential equation, and is usually approached using a numerical method such as the Euler Method. The Euler Method works by approximating the solution of the differential equation with a series of points. The points are calculated using the equation:V_{n+1} = V_n + \frac{\Delta t}{RC}(V_n - V_f + Ie^{(-V_n/n)})where V_n is the value of V at the current timestep, V_f is the final value of V, I is the current, n is the diode's ideality factor, and \Delta t is the time step size. By iteratively calculating the value of V at each time step, you can eventually find the solution to the equation.