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Questions/AnswersApril 14. Q. What topics will the Midterm cover? A. It will cover all topics that we discussed so far and those that we will discuss on Wednesday and Thursday. I have posted homework problems for this week so that you can see the topics that the Midterm will cover. Q. In the Midterm can we use the formula for the integrating factor for the first order linear ODE? A. Yes, you can. Q. Can we use calculators in the Midterm? A. No. We will choose problems so that computations can be done by hand. Q. Is it true that the Wronskian of two functions y1 and y2 is zero if and only if one of these two functions is a scalor multiple of the other. A. No. We have seen that if y1 or y2 is never zero, then your statement is true. Indeed, suppose y2 is never zero. Then the quotient y1 / y2 is constant if its derivative -W/(y2)2 is zero. But the derivative -W/(y2)2 is zero if and only if W=0. However, without assuming that y1 or y2 is never zero, your statement is not true. Example: Let y1(t)=t2. Let y2(t) be the function which is -t2 on the interval (-∞, 0) and which is t2 on the interval [0, ∞). Then the Wronskian is everywhere zero, but the functions y1 and y2 are not proportional. We will discuss this question more on Wednesday. Q. There are so many properties of Wronskian. Could you please formulate the important ones. A. OK, Suppose that y1 and y2 are two solutions of an ODE y''+ p(t)y' + q(t)y = 0. Let W(t) be the Wronskian of y1 and y2. Then 1) If the Wronksian W(t) is not zero at one point t0, then y1 and y2 form a fundamental set of solutions (this is the definition of a fundamental set of solutions). 2) If y1 and y2 form a fundamental set of solutions, then for any choice of initial conditions y(0)=y0 and y'(0)=y'0, there are constants c1 and c2 such that c1y1+c2y2 is the solution of the IVP y''+ p(t)y' + q(t)y = 0 y(0)=y0, y'(0)=y'0 3) If the Wronskian W(t) is not zero at one point t0, then it is not zero at any other point t. April 11. Q. When can we get our homework back. A. On Tuesday in class or during my office hours on Monday. Some misprints and comments were reported: L3 p6 -l8 dμ/dy should be dμ/dx L3 p7 l13 This Theorem is called the Existence and Uniqueness Theorem for first order ODE. We have also learned the Existence and Uniqueness Theorem for first order linear ODE (see L2 p4). Q. What are dx and dy in section 2.2 of the textbook? A. These are differential 1-forms. If you have taken a higher level course in mathematics, you may have learned about them. However, since some of you are not familiar with differential forms, we will not use them (they are not necessary for solving separable equations). |