System of Differential Equations

[confused student]

Homework Statement

(It should be noted that the actual problem has specific values associated with a, b, and c. However, at this point I'm trying to find a method to solve the problem rather than a specific solution).

Homework Equations

The Attempt at a Solution

When I was trying to solve this initially, I didn't realize that D changed with respect to t; for that reason my solution only applies for when D is constant. I calculated dv/dt with a constant D and multiplied that slope by small increments of t in order to get values of v. I know there are more efficient methods to do numerical integration like the Runge Kutta method, but I was trying to blow through the problem without them. My excel sheet started giving non real numbers when I added in the second differential equation (dD/dt), so I'm really stuck. How should I go about solving this with the additional differential equation? Thank you in advance for any assistance!

 

[BvU]

Hello and

started giving non real numbers

What does that look like, imaginary numbers ? Or just error indications ?

Can happen if D becomes negative. I would first try a smaller value for $a$ and/or a smaller step size

 

[confused student]

So I kept fiddling with it, and the problem was that D changed too quickly; I made my step size smaller and made my constants smaller. That generated actual numbers. One thing I'm still not sure of is my method to generate the plots. Does it make sense to multiply the slope by small increments of t if there is no t term in the differentials?

 

[BvU]

There is a $dt$ in the denominator, I should hope ?
What you do is you integrate: $f(t+\Delta t) = f(t) + f'(t) \Delta t$