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Unformatted text preview: McGILL UNIVERSITY
FACULTY OF SCIENCE FINAL EXAMINATION MATH 315 ORDINARY DIFFERENTIAL EQUATIONS Examiner: Professor J .J Xu _ Date: Friday April 13, 2007
Associate Examiner: Professor J. Labute Time: 2:00PM — 5:00PM INSTRUCTIONS This is a closed book exam
Answer all questions in the exam booklets provided.
Faculty standard calculators are permitted.
A table of Laplace Transforms has been provided. This exam comprises the cover, 3 pages of 8 questions and 1 page of tables) Fﬂ/ﬂ 7w M60 'SIJO) Final Examination MATH 315 Friday April 13, 2007 1. (10pts) Given the equation
2y(y + 23:2)da: + a:(4y + 33:2)dy = 0. (a) Show that this is not an exact equation, (b) Determine the values of the constants oz and [3, such that #(CL', y) = mayﬂ
is an integrating factor for this equation; (c) By using the integral factor found above, derive the general solution of the
equation. 2. (5pts) Perform the phase line analysis for the the following autonomous equa
tion: — = — 1 2 — 3 ,
, dt y(y ) (y )
and determine that 0 its equilibrium states; 0 the type of each equilibrium state, 0 the stability property of each equilibrium state, 0 sketch the integral curves in the physical plane (t, y), based on the above
phase line analysis without solving the equation. 3. (15pts) Find the general solution for the following equations:
(a) (D2 — 2D + 2)2(D2 — 1)y = 0; (b) (D2 + 4)y = 162: cos 23:; 4. (15pts)
(a) Find all values of a for which all solutions of 5
3323;” + omy' + Ey = 0 approach to zero as x —> oo. Final Examination MATH 315 Friday April 13, 2007 (b) Find the general solution for the following equations by using the method
of variation of parameters: 4 ﬂy” — 4m] + 6y : x sin 3:, (a: > 0). 5. (15pts) Given the following equation 1
9023;" + §(x + sin 2:)3/ + y = 0, (a) Find all the regular singular points (b) Derive the indicial equation and the exponents at the singularity for each
regular singular point; (c) Determine Whether the given equation has a solution that is bounded near
the regular singular point, has all solutions bounded near the regular sin
gular point, or has no non—zero solution bounded near the regular singular
point. 6. (10pts) Given the equation
2161/” + y’ + xy = 0,
(a) Show that a: = 0 is a regular singular point of the given equation and give the roots of the indicial equation; (b) Determine the recurrence formula for the coefﬁcients in the Frobenius series
expansion of the solution near :5 = 0; (c) Find at least the ﬁrst four terms of two linear independent solutions:
3/1 (x), 2120:) Final Examination MATH 315 Friday April 13, 2007 7. (10pts) (Choose one from two problems. You may get bonus points, if you
solved two.) Find the Laplace transform of the following functions: (a f (t) = 4cos2 bt, (b constant); (M
a ogtgl
ﬁn: a 1<tg2
a t>z 8. (10pts) (Choose one from two problems. You may get bonus points, if you
solved two.) Find the inverse Laplace transform of the following functions: (@ 2s + 3
F(8) = (s — 2)(s2 + 1)’
(b) I 2 —4s
m8) 2 358— 1' 9. (10pts) Solve the following IVP’S with the Laplace transform method: y" + 43) : sint — u27r(t) sin(t — 27r), y(0) = 0, y’(0) = 0. "r ...
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