Series Solution Coefficients for y'' - (sinx)y = cosx with Initial Conditions

In summary, the power series solution for the given differential equation is y = -5 + 3x + 0.5x^2 + (cosx + sinx)x^3/6 + (2cosx - sinx)x^4/24 + O(x^5). The coefficients for x^3 and x^4 can be found by expanding sin(x) and cos(x) in Taylor's series.
  • #1
JaysFan31

Homework Statement


Find the indicated coefficients of the power series solution about x = 0 of the differential equation:
y'' - (sinx)y = cosx, y(0) = -5, y'(0) = 3.
y = _ + _x + _x^2 + _x^3 + _x^4 + O(x^5)


Homework Equations




The Attempt at a Solution


This is going to be a tad confusing in typing it, but I hope it can be read.

I have the summation(anx^n)
This equals y.
y' = summation(nanx^n-1)
y'' = summation (n(n-1)anx^n-2)

Just to make it easier, we end up with
x^n[(n+2)(n+1)an+2-ansinx] = cosx
Thus, an+2 = (cosx + ansinx) / ((n+2)(n+1))

I know obviously that the first two terms (x^0 and x^1) are -5 and 3 respectively. I also know that the x^2 term is 0.5 by plugging in 0 for x. However, this doesn't work for the rest of them. I've done a lot of these types of problems, but this is the first one with sin(x) or cos(x), which puts an "x" in the an+2 equation (which I wrote above). What does x equal in this case? Can anyone just show me how to find the remaining coefficients because I'm pretty sure my equation is correct.
 
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  • #2
Expand sin(x) and cos(x) in Taylor's series.
 
  • #3
Yes, but I still don't have x do I?
 
  • #4
I got it. Thanks for the help.
 

FAQ: Series Solution Coefficients for y'' - (sinx)y = cosx with Initial Conditions

What is the general formula for finding the series solution coefficients for this differential equation?

The general formula for finding the series solution coefficients for this differential equation is given by:cn+2 = -cn/(n+2)(n+1), where c0 and c1 are the initial conditions for the solution.

How do you determine the initial conditions for the series solution?

The initial conditions for the series solution can be found by substituting the values of x and y into the given differential equation and solving for c0 and c1.

Can the series solution coefficients be negative?

Yes, the series solution coefficients can be negative. In fact, the general formula for finding the coefficients involves a negative sign.

Is there a limit to the number of terms that can be used in the series solution?

There is no specific limit to the number of terms that can be used in the series solution. However, the more terms you use, the more accurate your solution will be.

How do you know if the series solution is valid?

The series solution is valid if the series converges. This can be determined by using a convergence test, such as the ratio test or the root test. If the series converges, then the solution is valid and can be used to approximate the actual solution to the differential equation.

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