Lecture 3 Ordinary Differential equations Purpose of lecture: Solve 1 st order ODE by substitution and separation Solve 2 nd order homogeneous ODE Derive general and particular solutions of the above using examples of a LHO, unstable equilibrium, and a damped LHO.

Defining terms 1 st order means the highest differential contained within the equation is Ordinary differential equations have solutions that are a function of only one variable 2nd order means the highest differential contained within the equation is Homogeneous equations mean that in the equation below f(t) = 0 i.e. unforced

Defining terms The General solution is the broadest most long-winded version of the solution or The Particular solution is the result of applying the boundary conditions to the general solution Today we will learn how to solve any 1 st order and 2 nd order homogeneous ordinary differential equations……

1 st order homogeneous ODE 1 st method: Separation of variables e.g. radioactive decay gives 2 nd method: Trial solution Guess that trial solution looks like Substitute the trial solution into the ODE Comparison shows thatso write

1 st order homogeneous ODE This is the general solution The particular solution is found by applying boundary conditions e.g. At t=0 there are 100 nuclei; but at t=20 there were only 50 left So we can writeso The other boundary conditions tell us that So we can writeso The particular solution is therefore Have a go at example 1 in lecture 3 page 2

1 st order homogeneous ODE Example 1 in lecture 3 page 2 Example 1: The growth of an ant colony is proportional to the number of ants. If at t = 0 days there are only 2 ants, but after 20 days there are 15 ants, what is the general solution and what is the particular solution?

Step 3: General solution is or if m 1 =m 2 For complex roots solution is which is same as or Step 1: Let the trial solution be Now substitute this back into the ODE remembering that and This is now called the auxiliary equation 2 nd order homogeneous ODE Solving Step 2: Solve the auxiliary equation for and Step 4: Particular solution is found now by applying boundary conditions

2 nd order homogeneous ODE Let’s have a go now at the two most simple 2 nd order homogeneous ODEs Linear harmonic oscillator Unstable equilibrium

Step 4: When x = 0 t = 0 so so From general solution we find Conditions state when t = 0 and x = 0, velocity is V, so and so 2 nd order homogeneous ODE Step 1: Let the trial solution be So and Example 1: Linear harmonic oscillator with boundary conditions that it has velocity V when time t=0 at x=0 Step 2: The auxiliary is then and so roots are Step 3: General solution for complex is where  = 0 and   =  so Particular solution is therefore

Step 4: When t = 0 x = L so so From general solution we find Conditions state when t = 0 and x = L, velocity is 0, so so Step 2: The auxiliary is then and so roots are 2 nd order homogeneous ODE Step 1: Let the trial solution be So and Example 2: Unstable equilibrium, with boundary conditions that it has velocity 0 at time t = 0 and x(0) = L Step 3: General solution for real roots is

2 nd order homogeneous ODE Example 2: Unstable equilibrium Example 1: Linear harmonic oscillator

Extra example (not in notes) : Bike suspension consists of a spring in parallel with a damper unit. The ordinary differential equation defining motion is:- If find the general solution, plot x as a function of t, and categorise the motion as either under or over-damped? Damped harmonic oscillators Classical physics course tells us that turbulent drag force = kv 2 whereas laminar drag force = kv where k is a constant Although a function of viscosity, density, etc., laminar flow is favoured at low velocities such as in a damper So the effect of damper drag will be a function of

If then both m 1 and m 2 are negative so x(t) is the sum of two exponential decay terms and so tends pretty quickly to zero. The effect of the spring has been made of secondary importance to the huge damping and it is said to be over-damped, e.g. fire doors. If find the general solution, plot x as a function of t, and categorise the motion as either under or over-damped? Damped harmonic oscillators Auxiliary is roots are So and