Solving differential equation (involves power series)

In summary, the conversation discusses a problem involving a differential equation and finding a solution through the use of a series. The participants suggest different methods and techniques, such as using the Frobenius series technique and finding a Taylor series solution. They also discuss the potential issues that may arise with the chosen methods and suggest alternative approaches.
  • #1
EvilKermit
23
0
Hey guys. I'm new here. I've been trying to figure out how to solve this problem, and I'm still confused.

(-x^2 + 4x -3)* d2y/dx2 - 2(x-2) * dy/dx + 6y = 0

y(-2) = 1
dy/dx(-2) = 0

I set y = [tex]\sum[/tex]an(x+2)n (start at n=0, n goes to infinity)

dy/dx = [tex]\sum[/tex]ann(x+2)n-1 (start at n=0, n goes to infinity)

d2y/dx2 = [tex]\sum[/tex]ann(n-1)(x+2)n-2 (start at n=0, n goes to infinity


So:
0 = 6[tex]\sum[/tex]an(x+2)n - 2(x+2)[tex]\sum[/tex]ann(x+2)n-1 + (-x^2 + 4x -3)[tex]\sum[/tex]ann(n-1)(x+2)n-2

I can simplify this to:
0 = [tex]\sum[/tex]6an(x+2)n - [tex]\sum[/tex]2ann(x+2)n + (-x^2 + 4x -3)[tex]\sum[/tex]ann(n-1)*(x+2)n-2

This further simplifies to:

0 = [tex]\sum[/tex](6an-2ann)(x+2)n + (-x^2 + 4x -3)[tex]\sum[/tex]ann(n-1)*(x+2)n-2

However I don't know how to further simplify it, and where to go from here. Am I on the right track? I believe the answer will look like this:

y = a1+ a2x + a3x2 + a4x3....

If somebody could please help me out, I would very much appreciate it. Thank You
 
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  • #2
Look like you are on the right track but there is still a lot more to do.

I would prefer you start with let t=x+2 and replace all x in the DE with t-2.
 
  • #3
Your equation has regular singularities at x={1,3,[tex]\infty[/tex]}. Are these points within the range of the desired solution? If you need solutions that are near to these singularities you might have better luck using the Frobenius series technique and find solutions that cover your desired range. Start with the indicial equation.

Right now it looks to me like you're trying to find a Taylor series solution about the point where your initial conditions are defined (x=-2) but this is so close to the singular point (x=1) that you could have problems. However, it might be good enough if you're only concerned about solutions in the neighborhood of x=-2.
 
  • #4
How about set y = [tex]\sum[/tex]an(x)n (start at n=0, n goes to infinity)

Just do it the regular way! then you absorb the polynomial into the series. It is not as pretty, but you don't end up with the polynomial times a series.
 
  • #5
How about set y = [tex]\sum[/tex]an(x)n (start at n=0, n goes to infinity)

Just do it the regular way! then you absorb the polynomial into the series. It is not as pretty, but you don't end up with the polynomial times a series.
 

Related to Solving differential equation (involves power series)

1. What is a differential equation?

A differential equation is a mathematical equation that relates a function with its derivatives. It is used to model many physical phenomena in various fields, including physics, engineering, and economics.

2. What is a power series?

A power series is a series of the form ∑n=0∞ cn(x-a)n, where cn are constants, a is a fixed number, and (x-a)n are powers of x-a. It is a useful tool for approximating functions and solving differential equations.

3. How do you solve a differential equation using power series?

To solve a differential equation using power series, you first substitute the power series into the equation. Then, you equate the coefficients of each power of x and solve for the constants. Finally, you can use these constants to construct the solution to the differential equation.

4. What are the advantages of using power series to solve differential equations?

Power series provide a systematic and efficient method for finding solutions to differential equations. They can be used to solve a wide range of differential equations, including those that cannot be solved using other methods. Additionally, power series solutions can be used to approximate solutions to high accuracy.

5. Are there any limitations to using power series to solve differential equations?

While power series can be used to solve many types of differential equations, there are some limitations. Power series solutions may not exist or may be difficult to find for certain types of equations. Additionally, the process of finding the coefficients can be time-consuming and tedious for more complex equations.

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