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A Level Further Maths Pure 2 > Modelling With Differential Equations > Flashcards

Flashcards in Modelling With Differential Equations Deck (14)
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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

1. Make y the subject of the dx/dt equation and differentiate to find dy/dt
2. Substitute y and dy/dt found in 1 into the dx/dt equation for a second order differential equation in terms of x
3. Solve for x
4. Differentiate x and sub dx/dt and x into the dx/dt equation and solve for y


Derivative of dx/dt