MT1222: Calculus and Applications -- Part 1: ODEs

Corrections:

• Chapter 1: The function "g" in the existence and uniqueness statement should (of course) have been a "q".
• Example Sheet 2: In Question 5b, the initial condition should be applied at x=Pi, rather than at x=0! [Note that the ODE has a singular coefficient at x=0, so the existence and uniqueness theorem doesn't apply there. In fact, applying the IC y(0)=0 does not select a unique solution! [Things would be "worse" if we tried to assign a finite value at that point: The IVP comprising the ODE and the IC y(0)=1, say, has no solution!]
• Chapter 4: Correction to the modification of the method of undetermined coefficients for the case where a term in the ansatz solves the homogeneous ODE.
• Example Sheet 4: In Question 2(c), the ODE should be t^2 y'' - t y' + y = 0, and not t^2 y'' - 2 t y' + y = 0 [in the latter case the ansatz t^n does produce a second, linearly independent solution.]
• Example Sheet 6: In Question 1, I wrongly claimed that general fourth-order polynomials don't have a closed-form solution. That's a lie! The statement is true for fifth- and higher-order polynomials!

If you spot any other problems, please contact me by email (M.Heil@maths.man.ac.uk) or catch me after the lecture. Thanks.