MATH 202 Final, Spring 2015

American University of Beirut Math 202-Differential Equations Spring 2014-N. Nahlus, W. Raji, H. Yamani, M. Kobeissi, K. Aziziheris Final Exam- Time: 120 minutes May 14, 2014 Your Name:......................... and ID:................... Grades: 1/8 2/8 3/21 4/8 5/8 6/8 7/24 8/15 Total/100 Notes: • Calculators are not allowed. 1 PLEASE CIRCLE YOUR SECTION: Section 1(N. Nahlus, TTh 1230, Recitation F 1), Section 2(N. Nahlus, TTh 1230, Recitation F 12), Section 3(N. Nahlus, TTh 1230, Recitation F 8), Section 4(N. Nahlus, TTh 1230, Recitation F 9), Section 5(N. Nahlus, TTh 2 Recitation W 8), Section 6(N. Nahlus, TTh 2 Recitation W 12), Section 7(N. Nahlus, Th 2 Recitation W 11), Section 8(N. Nahlus, Th 2 Recitation W 2). Section 9(H. Yamani, MWF 10, Recitation Th 3:30), Section 10(H. Yamani, MWF 10, Recitation Th 11), Section 11(H. Yamani, MWF 10, Recitation Th 9:30), Section 12(H. Yamani, MWF 10, Recitation Th 8), Section 13(M. Kobeissi, MWF 2, Recitation F 12), Section 14(M. Kobeissi, MWF 2, Recitation F 3), Section 15(M. Kobeissi, MWF 2, Recitation F 4), Section 16(M. Kobeissi, MWF 2, Recitation F 5), Section 17(K. Aziziheris, MWF 8, Recitation M 2), Section 18(K. Aziziheris, MWF 8, Recitation F 2), Section 19(K. Aziziheris, MWF 8, Recitation F 4), Section 20(W. Raji, MWF 1, Recitation M 8), Section 21(W. Raji, MWF 1, Recitation M 9), Section 22(W. Raji, MWF 1, Recitation M 2), Section 23(W. Raji, MWF 1, Recitation M 5). 2 Usual Laplace formulas and rules: f(t) F(s) 1 1 s t 1 s2 tn n! sn+1 eat 1 s−a cos(kt) s s2+k2 sin(kt) k s2+k2 - L{f(t)} = � +∞ 0 e−stf(t)dt - L{f(n)t)} = snF(s) − sn−1f(0) − sn−2f′(0) − . . . − f(n−1)(0) - L{eatf(t)} = F(s − a) - L{f(t − a)U(t − a)} = e−asF(s) - L{U(t − a)} = e−as s - L{g(t)U(t − a)} = e−asL{g(t + a)} - L{tnf(t)} = (−1)n dn dsn F(s) - f ∗ g = � t 0 f(τ)g(t − τ)dτ - L{f ∗ g} = L{f}L{g} - L{δ(t − t0)} = e−st0 3 Problem 1. (8 pts) Use Gauss divergence Theorem carefully to find the outward flux � � S F · ndσ of the vector field −→ F = 1 � x2 + y2 + z2 (2x−→i + y−→j + 3z−→k � x2 + y2 + z2) across the surface ρ = 3. 4 Problem 2. (8 pts) Find the area of the surface cut form the plane x+2y +2z = 5 by the cylinder whose walls are x = y2 and x = 2−y2. 5 Problem 3. (21 pts, each part 7 pts) Solve the following DE’s: • xdy dx + 2y = x2y2, y(1) = 1/5 6 • dy dx = � 1 + (2x + y)3 (2x + y)2 − 2 7 • x2y′′ − 2xy′ + 2y = x3 √ 1 − x2 8 Problem 4. (8 pts) Use Frobenius method to find ONE series solution (corresponding to the larger root) of the equation x(x − 1)y′′ + 3y′ − 2y = 0 about the regular singular point x = 0. (Hint: Only find the first three non-zero terms) 9 Problem 5. (8 pts) Use Laplace Transform and solve the following DE: y′′ + 2y′ + 10y = δ(t − 2) + 13et, y(0) = 0, y′(0) = 13 10 Problem 6. (8 pts) Use Laplace Transform and solve the given equation: −3 8 (1 + t − f(t)) = � t 0 f(τ)(t − τ)3dτ 11 Problem 7. (24 pts, each part 8 pts) Answer the following ques- tions: 1. Find f(t) given that F(s) = ( s s2 + 25)(4), where the exponent (4) is the fourth derivative. 12 2. Use the definition of Laplace Transform to find the value of � ∞ 0 e−3t cos(4(t − 5))U(t − 5)dt 13 3. Find the Laplace of f(t) = 1 − cos t t 14 Problem 8. (15 pts) Find the general solution of the given system: X′ = AX, where A =   7 −4 0 1 0 2 0 2 7   15