MATH 202 Final, Spring 2008

AUB - Math department Prof. H. Gebran MATH 202 Hints for the final exam sample June 5, 2007 Exercise 6. Use the following facts 1) J′ 0 = −J1. 2) J0 is a solution to y′′ + 1 xy′ + y = 0. Exercise 7. There are several ways to solve the problem x′ = −x + 2y (1) y′ = −2x − y + e−t (2) x(0) = y(0) = 0. (3) First method (substitution-elimination). Equation (1) implies that y = 1 2 � x′ + x � and so y′ = 1 2 � x′′ + x′� . Replacing y and y′ by their expressions in equation (2) and simplifying you should get x′′ + 2x + 5x = 2e−t. Furthermore, by equation (3), we have that x′(0) = 0. I am sure now that most of you can solve the initial value problem x′′ + 2x + 5x = 2e−t x(0) = x′(0) = 0 by many ways. Second method (Laplace transforms). Applying the Laplace operator to equations (1) and (2) and using (3), you should get the algebraic system (s + 1)X − 2Y = 0 2X + (s + 1)Y = 1 s + 1, where X and Y are the Laplace transforms of x and y respectively. There are several ways to solve this system (Cramer’s, substitution, elimination). Learn this trick which may save time and avoid errors 1 (s + 1)[(s + 1)2 + 4] = 1 4 (s + 1)2 + 4 − (s + 1)2 (s + 1)[(s + 1)2 + 4] = 1 4 � 1 s + 1 − s + 1 (s + 1)2 + 4 � . Third method (matrix method). 1) Find the general solution of the homogeneous system by finding the eigenvalues and the eigenvectors of the matrix � −1 2 −2 −1 � . 2) Find a particular solution of the form e−t � a b � . Work out the details and compare your results. 1 Remark. In all the above methods you cannot escape from the polynomial r2 + 2r + 5 = (r + 1)2 + 4. Exercise 8. The integrals can be computed directly by parameterizing the boundary of the square (it is com- posed of four pieces). But if you recall Green’s theorem you are led to compute the divergence and the curl of F. Answer: zero. 2