MAP 4305 Practice Exam 3: Differential Equations - Prof. Patrick De Leenheer, Exams of Mathematics

Download MAP 4305 Practice Exam 3: Differential Equations - Prof. Patrick De Leenheer and more Exams Mathematics in PDF only on Docsity! Practice Exam 3: MAP 4305∗ 1. Does 5xy′′ + 4(1− x2)y′ + y = 0, x > 0, have a solution which is bounded near zero? Notice that to answer this question, you only need to consider the indicial equation. 2. Determine the form of a series expansion about x = 0 of 2 linearly independent solutions to: x2y′′ − xy′ + (1− x2)y = 0, x > 0. Do not find a recursion formula for the coefficients. 3. Find the first three non-zero terms in a series expansion about x = 0 of 2 linearly independent solutions to: 3xy′′ + (2− x)y′ − y = 0, x > 0. 4. Draw solutions in the (x, y) plane of the following system in polar coordinates: ṙ = sin r θ̇ = −1 Are there any non-trivial periodic solutions? If yes, are they limit cycles? If there are non-trivial periodic solutions, how many are there, and what can be said about their stability? 5. The Legendre polynomials Pn(x) satisfy the following recurrence relation: (n+ 1)Pn+1(x) = (2n+ 1)xPn(x)− nPn−1(x). Given that P0(x) = 1 and P1(x) = x, determine P2(x), P3(x) and P4(x). ∗Instructor: Patrick De Leenheer. 1