Power Series Method: Solve y'' + 2xy' + 2y = 0

In summary, the conversation is about using the power series method to find the general solution for the equation y'' + 2xy' + 2y = 0. The person discussing the problem has attempted to solve it by substituting y(x) with a power series and equating the constant and linear terms with the original equation. They ask for help and are advised to use LaTeX to write the sums.
  • #1
Fairy111
73
0

Homework Statement



Use the power series method to find the general solution to:

y'' + 2xy' + 2y = 0

Homework Equations





The Attempt at a Solution



I let y(x)=a0 + a1x + a2x^2 +...+anx^n + ...

Then i found out what y'(x) and y''(x) was. I then equated the constant terms with the original eqn and tried to then do the same with the linear terms. I am not really sure what I am doing though and how to continue.
 
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  • #2
Why don't you show what you've got.

It will probably be easiest if you use LaTeX to write the sums...For example, If you write:

[$tex] y(x)=\sum_{n=0}^{\infty} a_n x^n[$/tex]

And delete the $ signs , you get this:

[tex] y(x) =\sum_{n=0}^{\infty} a_n x^n [/tex]
 

1. What is the power series method?

The power series method is a mathematical technique used to solve differential equations. It involves representing the unknown function as a series of terms, each with a different power of the independent variable.

2. How is the power series method used to solve differential equations?

The power series method involves substituting the series representation of the unknown function into the given differential equation, and then solving for the coefficients of the series using algebraic manipulation and the recursive relationship between the coefficients.

3. What is the advantage of using the power series method?

The power series method allows for the solution of differential equations that cannot be solved using other methods, such as separation of variables or the method of undetermined coefficients. It also provides an exact solution, rather than an approximation.

4. Can the power series method be used for all types of differential equations?

No, the power series method is most commonly used for linear differential equations with variable coefficients, such as the one given in the question. It may not be applicable for non-linear or partial differential equations.

5. Are there any limitations to the power series method?

One limitation of the power series method is that it can be time-consuming and tedious, especially for equations with high-order derivatives. It also requires the initial conditions or boundary conditions to be known in order to determine the coefficients of the series.

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