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Differential Equations Comprehensive Exam Spring 2020, Schemes and Mind Maps of Differential Equations

Duke UniversityDifferential Equations

Differential Equations Comprehensive Exam. Spring 2020. Student Number: Instructions: Complete 5 of the 8 problems, and circle their numbers below – the ...

Typology: Schemes and Mind Maps

2022/2023

Uploaded on 05/11/2023

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Differential Equations Comprehensive Exam
Spring 2020
Student Number:
Instructions: Complete 5 of the 8 problems, and circle their numbers below – the uncircled
problems will not be graded.
1 2 3 4 5 6 7 8
Write only on the front side of the solution pages. A complete solution of a problem
is preferable to partial progress on several problems.
School of Math Georgia Tech
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Download Differential Equations Comprehensive Exam Spring 2020 and more Schemes and Mind Maps Differential Equations in PDF only on Docsity!

Student Number: Instructions:problems willWriteis preferable to partial progress on several problems. only on the front sideDifferential Equations Comprehensive Exam Complete 5 of the 8 problems, andnot 1 be graded. 2 of the solution pages. A 3 Spring 2020 4 circle 5 their numbers below – the uncircled 6 complete solution 7 8 of a problem

School of Math Georgia Tech

  • Differential Equations Comp1. Find all critical points of ¨2. Consider ¨3. Can you find a periodic solution of the system4. Consider the equation ˙a sink, source, saddle, center, or something else in the full nonlinear system.why?How aboutExplain. and constantq + ˙ qa, b + (. Is there an unbounded solution asq − 1)x 5 = x= 0. Does lim + sinA A((t{t)) =xxy ˙ ˙x with {= 0. For each of them, demonstrate rigorously if it is [== 5 xy ˙ ˙/ y− 3 x −xt==− →∞∈a − xy− cosR 5 y−x q 2 + (sin ( + cos t−txand ) exist? If yes, what is the limit? If not, 5 y 1 / 3 + sint t?) 5 bt →? +t ]∞? Spring
      1. Consider the following initial value problem6. Let7. Let Ω be an open bounded subset in Find the time when the solution blows up first. u u where x ( Find the function∈∈r, θ CΩ. Prove thatD) = 1 + 2 (0g, 1 (, r(Ωx)) be the disk on ×≥ cos(0 0, and there is a finite positive number, 2 ∞( θu)) solves the following) on (ux, t(ρ, θ C) ≥.) in polar coordinates so that it is harmonic on { R0 for all ( uuu 2 t(( x, tx,−uucentered at the origin with radiust( 0) = ∆x,) = 0+u 0) =uu =x, t g x , x(Rf x= 0 ) (1 + n)x ∈, f or x ∈) 1 with smooth boundary, xu, x (Ω ∂xΩ 2 ×∈, t >∈ ∈ (0 R∈C R,, t > 1 Ω∞ (n 0 R, t >,,M).) 0 .,such that 0 , r ∂|fwith boundaryΩ. (x )| ≤Assume that D (0M, r for all) and C