MATH 251Spring 2003Midterm Exam I1. (8 points) Classify the following differential equations as linear or non-linear and state their order.linear/ ordernon-linearln(t)d2ydt2+ 3etdydt− y sin t = 02y0− y2= ety000+ (t2− 1)y + cos t = 0y00− sin(t + y)y0+ (t2+ 1)y = 02. (6 points) What is the integrating factor of the differntial equationt2y00− 4ty = et− cos 2t(a) µ(t) = 1/t4(b) µ(t) = e4 ln t(c) µ(t) = −t4(d) µ(t) = −2t23. (6 points) The Existence and Uniqueness Theorem guarantees that the solution to(t + 2)y00− sin ty0+tyt − 4=e2tt, y(−1) = 0(a) is valid on (−∞, ∞)(b) is valid on (−π, 0)(c) is valid on (−2, 0)(d) The solution does not exist4. (6 points) Which of the equations below has y(t) = c1et+ c2e−2tas its general solution?(a) y0− 2y = 0(b) y00− y0− 2y = 0(c) 2y00+ 2y0− 4y = 0(d) y00− 3y0− 2y = 05. (6 points) What is the form of the particular solution ofy00+ 4y0+ 4y = te−2t+ 2t2cos 2t − 3(a) y(x) = (At3+ Bt2)e−2t+ (Ct2+ Dt + E) cos 2t(b) y(x) = (At2+ Bt)e−2t+ (Ct2+ Dt + E) cos 2t + F(c) y(x) = (At + B)e−2t+ (Ct2+ Dt + E) cos 2t + (F t2+ Gt + H) sin 2t + I(d) y(x) = (At3+ Bt2)e−2t+ (Ct2+ Dt + E) cos 2t + (Ft2+ Gt + H) sin 2t + I6. (10 points) Let y0= −y(y2− 4).(a) Find all equilibrium solutions.(b) Determine the stability of each equilibrium solution. Justify your answer.(c) If y(5) = 1, what is limt→∞y(t)?MATH 251 Midterm Exam I PAGE 27. (14 points) Certain bacteria has a volume of 2nl which we assume to be mostly cytoplasm. This bacteriais placed in a substance poluted with 3ng/nl of certain harmful chemical. The bacteria exchanges fluidswith its media at a rate of 1/100nl/s, that is there is a flow of 1/100nl/s of the poluted substance intothe bacteria, and the (well mixed) cytoplasm flows out of the bacteria at the same rate. Assume that thebacteria is initially clean from the chemical.(a) Find a formula for the amount of the chemical in the cell at any given time.(b) The bacteria is going to die when the concentration of the chemical reaches 3ng/nl. When is thisgoing to happen?8. (10 points) Consider the differential equation:(ln(y) + 2xy + 2 cos 2x) + (xy+ x2)y0= 0(a) Show that the above equation is exact.(b) Give the general solution to the equation.9. (12 points) (a) Are the functions y1(t) = t3and y2(t) = 1/t solutions to the differential equationt2y00− ty0− 3y = 0(b) Are the functions y1and y2linear independent? Justify your answer.(c) What is the general solution of the differential equation of part (a). Justify.10. (12 points) (a) Find the solution to the homogeneous differential equationy00+ 3y0− 4y = 0(b) Find the general solution toy00+ 3y0− 4y = 50 sin 2t11. (10 points) Solve the initial value problemy00+ 6y0+ 9y = 0, y(0) = 3, y0(0) = −1ANSWERS:1. i. linear, order 2; ii. nonlinear, order 1; iii. linear, order 3; iv. nonlinear, order 2.2. (a); 3. (c); 4. (c); 5. (d).6. (a) equilibrium solutions: y = −2, 0, 2; (b) y = −2 asymptotically stable; y = 0 unstable; y = 2asymptotically stable; (c) y(t) = 2.7. (a) Q(t) = −6e−t/200+ 6; (b) it will never happen (or, when t approaches infinity).8. (b) x ln y + x2y + sin 2x = C.9. (a) yes; (b) yes: W (y1, y2) = −4t, which is not equal to zero when t is not zero; (c) since y1and y2aretwo linearly independent solutions, the general solution is therefore y(t) = C1t3+ C2t−1.10. (a) y(t) = C1e−4t+ C2et; (b) y(t) = C1e−4t+ C2et− 3 cos 2t − 4 sin 2t.11. y(t) = 3e−3t+ 8te−3t.Page
View Full Document
Unlocking...