Differential Equations – Day 3

Model: Mixing Problem – 1 Tank

Rum and Coke solution.

Assumptions:

1. Maintain constant tank volume.

2. Inlet and outlets supply and drain at the same rate.

3. Instantaneous & thorough mixing

How much rum is present in the tank at any time t?

We want to track the amount of rum at a given time.

Rate in =

Rate out =

More on Separable Equations

Exact Equation of the form:

Where M and N are expressed entirely in terms of their paired derivative we are able to find …

When f(x,y)= 0, it follows that

If With is a differential of then the equation is an exact equation which means is an implicit solution of the original equation.

This comes out of the mixed partial work that we did in Math 1D. If the mixed partials are equal, the equation is exact. Meaning –

Example:

This is an exact solution

Super position principle & Variation of Parameter

we would like to solve for . The solution to the second is called the complementary function, called . The solution to the first is where is a particular solution for equation and compensates for the input function.

So our original equation can be rewritten as orwhere for our homogeneous solution.

where M(x) is a separable equation. Solve the homogeneous solution first then allow the form of the homogeneous equation to help you get going on the right track for the particular/non-homogeneous. Multiply by M(x) and solve for M(x). Do remember to check things out after the fact though.

Solve

Since My and Nx are not the same, we can do some fancy work to make things fun.

We can use this to solve our equation for f and find out what our integrating factor is.

Assume that Fy or Fxis 0 and solve for f=F(x) or f = F(y), respectfully.