Power Series Approximation of an IVP

In summary, the conversation discusses finding the first four nonzero terms of the power series approximation of the solution to the non-homogeneous differential equation y"-4y=4t-8e^-2t with initial conditions y(0)=1 and y'(0)=-1. The solution is expressed as y=\suma_n*t^n where the summation goes from 0 to infinity. The speaker has successfully found the first four terms for the homogeneous problem, but is unsure how to proceed with the non-homogeneous equation. They have recursive formulas for the coefficients, but most of them end up being equal to 0. They ask for clarification on how to derive the formulas for the coefficients.
  • #1
LUMath2012
1
0
1. Find the first four nonzero terms of the power series approximation of the solution.
y"-4y = 4t-8e-2t y(0)=1, y'(0)=-1


2. y=[tex]\sum[/tex]a_n*t^n where the summation goes from 0 to infinity



3. I have done a homogeneous problem similar to this and had no problems finding the first four terms. However, I am confused as far as where to go with the non-homogeneous equation. I have recursive formulas for the first four terms but all of them but one end up being equal to 0. Just wondering where I go from here.
 
Last edited:
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  • #2
how did you derive your formulas for the coefficients?
 

1. What is a power series approximation?

A power series approximation is a mathematical method used to approximate the solution to an initial value problem (IVP). It involves representing the solution as a series of terms, with each term containing a power of the independent variable.

2. How is a power series approximation used in solving an IVP?

A power series approximation is used in solving an IVP by substituting the series into the differential equation and solving for the coefficients. This results in an infinite series that approximates the solution to the IVP.

3. What are the advantages of using a power series approximation?

The advantages of using a power series approximation include its ability to provide a more accurate solution compared to other approximation methods, its versatility in solving a wide range of differential equations, and its ability to handle non-analytic functions.

4. What are the limitations of a power series approximation?

One limitation of a power series approximation is that it can only approximate the solution to an IVP within a certain interval of convergence. If the initial conditions of the IVP fall outside of this interval, the approximation may not be accurate. Additionally, the convergence of the series may be slow, making it impractical for certain problems.

5. How can the accuracy of a power series approximation be improved?

The accuracy of a power series approximation can be improved by including more terms in the series, which will result in a higher degree polynomial and a more accurate solution. Additionally, using a smaller step size in the numerical method used to solve the series can also improve accuracy.

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