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Ted123
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Homework Statement
[PLAIN]http://img220.imageshack.us/img220/7427/diff5.jpg
The Attempt at a Solution
Done (a). How do I go about (b) and (c)?
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Mark44 said:For b, to construct the equation y'' + y = 0, you must have p(x) [itex]\equiv[/itex] 0 and q(x) [itex]\equiv[/itex] 1. What implication does this have on what you established in part a?
Linear independence refers to a set of vectors that are not dependent on each other, meaning that one vector cannot be written as a linear combination of the others. In other words, each vector in the set adds unique information and cannot be duplicated by a combination of the other vectors.
In differential equations, linear independence is crucial because it allows us to create a set of linearly independent solutions that can be used to form a general solution. This general solution can then be used to solve more complex differential equations, making the process more efficient and accurate.
A set of vectors is linearly independent if the only solution to the equation c1v1 + c2v2 + ... + cnvn = 0, where c1, c2, ..., cn are constants and v1, v2, ..., vn are the vectors, is c1 = c2 = ... = cn = 0. In other words, the only way for the linear combination of the vectors to equal 0 is if all of the constants are 0.
No, a set of linearly dependent vectors cannot be used to form a general solution in differential equations. This is because the linearly dependent vectors would provide redundant information and therefore would not be able to accurately solve the differential equation.
In a differential equation of order n, there must be n linearly independent solutions in order to form a general solution. This is because each solution adds unique information to the overall solution, and the number of unique solutions needed is equal to the order of the differential equation.