Modelling With Differential Equations Flashcards
Setting up filling a container questions
Volume in * concentration in - (volume out * concentration of liquid in container)
Simple Harmonic Motion
A motion in which the acceleration is always towards a fixed point O and is proportional to the displacement from O
d^2x/dt^2 formulae
ω^2x or v(dv/dx)
SHM finding an equation for v from acceleration
-ω^2x = v(dv/dx) and separate variables
SHM finding an equation for x
Use d^2x/dt^2 + ω^2 x = 0. You can then use the CF to find an equation for x, use R formulae or sub into x and a differentiated equation for v to find boundary conditions depending on the information given
Damping method
A damping force proportional to velocity may be added and solve as a homogenous equation
Damping distinct roots
x = Ae^-αt + Be^-βt
Heavy damping, no oscillation
Damping equal roots
x = (A + Bt)e^-αt
Critical damping, the limit at which no oscillation occurs
The graph sees the displacement rise first and then curve tending to 0
Damping complex roots
x = e^-kt(Acos(αt) + Bsin(βt))
Light damping, the amplitude gradually reduces
Forming damped/forced differential equations
You may need to use F=ma and resolve forces on the object, setting in the opposite direction to the acceleration
Forced and damped harmonic motion method
d^2x/dt^2 + k dx/dt + ω^2 x = f(t)
Solve as a non-homogenous differential equation
Showing what happens at t gets large
The e^-kt section –> 0 so use the other section, potentially using R formulae
Coupled method to solve for x
- Make y the subject of the dx/dt equation and differentiate to find dy/dt
- Substitute y and dy/dt found in 1 into the dx/dt equation for a second order differential equation in terms of x
- Solve for x
- Differentiate x and sub dx/dt and x into the dx/dt equation and solve for y
Derivative of dx/dt
d^2x/dt^2
