Hard power series and initial value problem question

[calculusisrad]

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 (a_n)(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 a_n is related to a_n-1 for n >1. From this, find an explicit formula for a_n.
Finally, use the known power series representation for e^x to conclude that y(x) = Ae^x

The Attempt at a Solution

I know a_n = 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!

 

[jackmell]

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 (a_n)(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 a_n is related to a_n-1 for n >1. From this, find an explicit formula for a_n.
Finally, use the known power series representation for e^x to conclude that y(x) = Ae^x

The Attempt at a Solution

I know a_n = 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!

If the solution can be represented as:

$$y(x)=a_0+a_1 x+a_2 x^2+\cdots$$

and you know y(0)=A, then you know what a_0 is then and when you take the derivative of y(x), it's

$$y'(x)=a_1+2a_2 x+3 a_3x^3+\cdots$$

and since you're given y'=y, then what about equating the respective power series for y'=y then equating coefficients? Doing that, can't you find an explicit expression for a_n? I'll do two:

$$a_1=A$$
$$2a_2=a_1$$

but since a_1=A then $a_2=\frac{a_1}{A}$. Now do a few more and notice the trend. Then come up with the general expression for a_n.