Lecture 5 Inhomogeneous ODEs Intro to Fourier Series You’ll solve an inhomogeneous ODE extra example We’ll look at using complex numbers to solve IODEs Introduction to Fourier analysis Today I’m now setting homework 1 for submission 31/10/08

Extra example of inhomo ODE Solve Step 1: With trial solution find auxiliary is Step 2: So treating it as a homoODE Step 3: Complementary solution is Step 4: Use the trial solution and substitute it in FULL expression. so cancelling Comparing sides gives…. Solving gives Step 5: General solution is

Finding partial solution to inhomogeneous ODE using complex form Sometimes it’s easier to use complex numbers rather than messy algebra Since we can write then we can also say that and where Re and Im refer to the real and imaginary coefficients of the complex function. Let’s look again at example 4 of lecture 4 notes Let’s solve the DIFFERENT inhomo ODE If we solve for X(t) and take only the real coefficient then this will be a solution for x(t) Sustituting so Therefore since take real part

Find full solution of inhomoODE extra example below Step 3: Complementary solution is Step1 and 2: For trial of auxiliary is roots Step 4: Use the trial solution and substitute it in FULL expression. so cancelling Comparing sides gives…. Solving gives so Step 5: General solution is

Find partial solution to example of inhomoODE using complex form Complementary solution is unchanged so..... Is it faster this way? Step 4: Imagine equation we want to solve is of the form So we pick a trial solution Substituting into FULL equation gives Cancelling gives so and Step 5: Now so We therefore can write and if we set So since Particular Solution is