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Exam3Spring94 - LABORATORY TEST I3 65.2400 ~ 07 08 09 April...

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Unformatted text preview: LABORATORY TEST I3 65.2400 ~ 07. 08. 09 April 25. 1994 NAME -____.____-~____._______ SECTION ma 1.Consider the system I’ = -I + 2y. 3/ = (1/2)3: —— 3y. (a) Draw a direction field on the square —2 g r 5 2, —2 g y g 2. From an inspection of the direction field classify the origin as to type saddle point node spiral point center and whether it is asymptotically stable stable unstable (b) Plot the solution that passes through the point (1. 2) and sketch its graph on the axes below. “3 (C) For the solution in part (b) sketch the graphs of I versus t and y versus t. for O S t < 5. on the axes below. Put scales on the axes and label the graphs clearly. _ L3 (d) For large t what is the approximate ratio of I to y for this solution? ratio = ‘2. Consider the system 13’: 9. y’ = I - (ll/(3)1J —1‘1/4‘y. (a) Draw a direction field on the square —-1 _<_ r g 4. —-4 S y S «1. By looking at the direction field determine the approximate location of each critical point (equilibrium solution). (b) Classify each critical point as to type (saddle point. node. ...) and state whether it is asymptotically stable, stable. or unstable. 'n (c) Plot the trajectories through the points (0.2) and (0.3) for 0 S t S 20. Use a stepsize of 0.1 to obtain smooth curves. Sketch the trajectories on the axes below. ‘1 (d) There is a trajectory between the ones in part (c) that behaves in a different way from both of them. What happens to this trajectory as t —~ 00? 65.2400 - O7. 08. 09 TEST 3 April 27. 1994 Name Section 07 08 09 Please answer all questions, showing your work in detail. You may refer to one sheet (8.5 by 11 inches) of notes. No other notes, books, references, calculators, etc. are permitted. Problems 3 and 4 count 20 points each; problem 5 counts 30 points. "' "MW NAME -_______.__-.___.___..-____ SECTION _____..-___.__-_-_- 3 (a) Consider the partial differential equation uxz+uyy+4u=0 Assume that u(:I:, y) = X (I)Y(y) and find ordinary differential equations satisfied by X(:c) and Y(y). _: l (b) A system of equations 93’ = Act has the phase portrait shown below. What can be said about the eigenvalues and eigenvectors of A? NAME _...._______ ______ __. SECTION ______.. _____ _ 4. The eigenvalues and ,eigenvectors of a certain matrix A are 1 =—1+2', (I): - n z E 3+42‘ ’ 1 =—1- ', (2) = . T2 21 5 (3—4,) (a) Find the general solution of x’ = AI in terms of real-valued functions. (b) Also find the solution that satisfies the initial conditions NAME .__.______-_--_---___-____-- SECTION _____________________ 5. Consider the system I, = y, y’ = I - (1/6)133 — (1/4)?!- (a) Find all critical points (equilibrium solutions). (b) Find the linear system that approximates the given system near the origin. Based on the linear system determine whether the origin is asymptotically stable, stable, or unstable. Also classify it as to type (saddle point, node,...). CONTINUED NEXT PAGE... (C) Choose one of the critical points that is different from the origin. Find the approximate linear system valid near this critical point. Based on the linear system determine whether this critical point is asymptotically stable. stable. or unstable. Also classify it as to type (saddle point. node....)_ (d) One of the trajectories of this system is shown below. 0'1; the other set of axes sketch the graph of 1‘ versus t for this trajectory. ‘1 ...
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