Solving second order differential equations Welcome to the

Solving second order differential equations Welcome to the session! • Please use the chat to say hi and let us know which department you are from • Please keep your microphone and video switched off. If you have any questions, ask them in the chat or use the raise hand button • Make sure to sign in with your full name (or a nickname and your ucard number) so that we can send any relevant resources after the session While you are waiting: What can be modelled using second order ODEs in real life?

Warm-up exercise Some applications • Mechanics: simple harmonic motion/vibrations • Electrical: circuits • Quantum mechanics: particle energy equations

Goals for today • Recap methods for solving homogeneous and inhomogeneous linear second order ODEs • Refresh the different types of solutions • Solve problems involving boundary or initial conditions

General form •

Homogeneous linear second order ODE •

Technique •

Technique •

Technique •

Technique •

Technique Homogeneous case – an example

Inhomogeneous linear second order ODE •

Technique •

Technique Proof for particular solution technique

Technique • Function type Guess A constant A finite product of the above The product of their respect guesses

Technique •

Technique Inhomogeneous case – an example

Technique Boundary/initial conditions Boundary conditions allow us to find the value of constants in our general solution. For a second order ODE, you will usually need two conditions!

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