- #1
calculusisrad
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Homework Statement
We know that y = Aex is the solution to the initial value problem dy/dx = y; y(0) = A.
This can be shown by solving the equation directly. The goal of this problem is to reach the same conclusion using power series.
Method: Let y be a solution to the initial value problem, and suppose y has the
power series representation
y(x) =the sum from n= 0 to n= infinity of (an)(x^n)
which equals a0 + a1x + a2x^2 + ...
First finnd a0. Next, take the term-by-term deriviative of the series to fi nd a power
series representation for dy/dx . Using the fact that dy/dx = y, obtain a formula which
shows how an is related to an-1 for n >1. From this, find an explicit formula for an.
Finally, use the known power series representation for e^x to conclude that y(x) = Ae^x
The Attempt at a Solution
I know an = f(n)(0)/n!
And I know the function is a power series centered at 0.
but I don't really know where to go from there?
i just don't know where to start on this. please help. if I see how to get started, I will be able to understand. I did try, but I don't know what to do!