Math 54, Summer 2009, Lecture 4Differential Equations ReviewThe purpse of this sheet is to outline some of the major types of problems we covered in ourstudy of differential equations. This is a way to start your review, but for a more completepicture it would be best to go back through old homework problems.Find a formula for the general solution to:• ODEs of the form ay00+ by0+ c = 0 (Sections 4.2-4.3)• ODEs of the form ay00+ by0+ c = f(x) for certain kinds of functions f(x). Use themethod of undetermined coefficients and the superposition principle (Sections 4.4-4.5)• ODEs of the form any(n)+ an−1y(n−1)+ · · · + a1y0+ a0y = 0. This requires findingthe roots of the polynomial anrn+ · · · + a0, which will either be given to you or canbe found with guess/check and polynomial long division (Section 6.2). You should beable to handle real and complex roots of this polynomial.• systems of first-order ODEs with constant coefficients of the form ~x0= A~x (Sections9.5-9.6). You should be able to handle complex eigenvalues/eigenvectors.• heat flow problems ut= βuxx, with initial condition u(x, 0) = f(x) and homogenousboundary values for u (u(0, t) = u(L, t) = 0)Any of the ODEs listed above could come with initial or boundary conditions, which onewould then plug into the general solutions and solve for the appropriate values of the pa-rameters.Solvability of initial value problems without a formula:• Solvability of initial value problems for linear ODEs with continuous data (Section 6.1)• Solvability of initial value problems for systems of first-order linear ODEs with contin-uous data (Section 9.4)1You should also know how to:• transform systems of linear ODEs and higher order ODEs into a matrix system innormal form (Section 9.1)• compute eAand eAtwhen A is diagonalizable, and work with the basic properties ofthe matrix exponential (coming from the series definition, or from the interpretationas a fundemental matrix) (Section 9.8)• compute the Fourier series of a function whose domain is [−T, T ], and the Fourier sineand cosine series of a function whose domain is [0, T ] (Sections 10.3-10.4)• graph and write a formula for the function a Fourier series converges to (Sections10.3-10.4)• use separation of variables to turn a PDE into a system of ODEs, and then turn theseODEs into a set of fundamental solutions (Section
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