Connect and share knowledge within a single location that is structured and easy to search. I'm currently going through Harvard's Abstract Algebra using Michael Artin's book, and have no real way of verifying my proofs, and was hoping to make sure that my proof was right. Your idea of evaluating at particular points is a good one, and they build on each other. Differentiation can be powerful in cases like this. BobaFret's suggestion of using the Wronskian is one possibility, but you can do it more directly.
Your errors have been discussed well in Jonas Meyer's answer, and he also discussed differentiation. I want to add why differentiation is a good idea. This process of "looking at a condition near a point" is often means differentiating the condition there. It is often easier to differentiate a given condition at an easy point than to study the condition elsewhere.
Even if there are several easy points you could pick, it can be beneficial to choose only one but look at different derivatives at the same point. This is not just a little calculus trick; this idea is not at all unusual in my research. Do you know what the Wronskian is? Sign up to join this community. The best answers are voted up and rise to the top.
Stack Overflow for Teams — Collaborate and share knowledge with a private group. Create a free Team What is Teams? So the idea of a linear combination of two functions is this: Multiply the functions by whatever constants you wish; then add the products. Show that y 3 is a linear combination of y 1 and y 2. If the answer were yes, then there would be constants c 1 and c 2 such that the equation. One more definition: Two functions y 1 and y 2 are said to be linearly independent if neither function is a constant multiple of the other.
Checking that two functions are dependent is easy; checking that they're independent takes a little more work. If they weren't, then y 1 would be a constant multiple of y 2 ; that is, the equation. This is where the Wronskian can help. Be very careful with this fact. In fact, it is possible for two linearly independent functions to have a zero Wronskian! This fact is used to quickly identify linearly independent functions and functions that are liable to be linearly dependent.
In this case if we compute the Wronskian of the two functions we should get zero since we have already determined that these functions are linearly dependent.
Here we know that the two functions are linearly independent and so we should get a non-zero Wronskian. This is not a problem. If its non-zero then we will know that the two functions are linearly independent and if its zero then we can be pretty sure that they are linearly dependent. Now, this does not say that the two functions are linearly dependent! However, we can guess that they probably are linearly dependent.
Therefore, the functions are linearly dependent. So, this means that two linearly dependent functions can be written in such a way that one is nothing more than a constants time the other. Go back and look at both of the sets of linearly dependent functions that we wrote down and you will see that this is true for both of them. Write down the following equation.
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