Differential equations (typical c4 test question)

[thomas49th]

Homework Statement

Liquid is pouring into a container at a constant rate of 20 cm³/s and is leaking out at a rate proportional to the volume of liquid already in the container

a) Explain why, at t seconds, the volume, V cm³, of liquid in the conainer satisfies the differential equation:

$$\frac{Dv}{dt} = 20 - kV$$
where k is a positive constant

The container is initially empty

b) By solving the differential equation show that

$$V = A + Be^{-kt}$$

giving the values of A and B in terms of k

The Attempt at a Solution

a) Well the change in volume differs with time. 20 cm³/s is pouring in the the container minus a proportional rate of the volume in the container at the time.

b) NOT SURE

err do i have to do some flipping of differentials around?

Thanks :)

 

[nicksauce]

b) yes... dv / (20-kV) = dt, then integrate and solve for V(t)

 

[Defennder]

In other words, only separation of variables is needed to solve this DE.