<p> Math 44 – Differential Equations Saturday, March 10, 2007 Mini quiz 3 - SOLUTIONS</p><p>1. (5 pts) Find all solutions (that is, the “general solution”) to this system: x'( t )= 4 x ( t ) + 2 y ( t ) y'( t )= 3 x ( t ) - y ( t ) 4 2 The matrix is A = , its characteristic equation is 2 , and its eigenvalues (3- 1) l-3 l + 2 = 0 2 1 are 1 = 5 and 2 = –2 with corresponding eigenvectors and . So the general solution is this: (1) (-3) 5t- 2 t x( t )= 2 c1 e + c 2 e 5t- 2 t y( t )= c1 e - 3 c 2 e There are several equivalent ways to describe this family of solutions. Scoring: 2/5 for eigenvalues and eigenvectors, max 3/5 for wrong answer, no matter how trivial the error, because the solutions can be checked.</p><p>2. (4 pts) One solution to a certain system (in complex numbers) is It doesn’t matter, but (2+i ) t the system is x( t ) = e x' = x - y y( t )= (1 + i ) e(2+i ) t { y'= 2 x + 3 y Find at least one solution involving only real numbers. x( t ) = y( t ) =</p><p>(2+i ) t 2 t Intended solution: e= e(cos t + i sin t ) , so the given solution becomes 2t轾 2 t 轾 2 t x( t )= e( cos t + i sin t) =臌 e cos t + i 臌 e sin t 2t轾 2 t 轾 2 t y( t )= (1 + i ) e( cos t + i sin t) =臌 e (cos t - sin t ) + i 臌 e (cos t + sin t ) To get a real solution, just strip out the real part of this. 轾2t x( t )= 臌 e cos t 轾2t y( t )=臌 e (cos t - sin t ) Or one might strip out the imaginary part. There are many alternative solutions.</p><p>Did you notice that the given “solution” doesn’t actually satisfy the system shown? Sorry; I flipped the sign of y. Full credit for any nontrivial real solution that (a) follows from the given “solution” or (b) actually solves the system shown. Compromise 2/4 for the surprisingly popular “x(t)  0, y(t)  0.”</p><p>3. (1 pt) If you have time, sketch a vector field OR a direction field OR a bunch of solution curves for the system in problem 1. Full credit for a picture that shows two “eigenlines” and is clearly a saddle. (end)</p>