# Differential equation with power series method

• Schfra
In summary, the conversation revolved around solving the differential equation y' = x^2y using the power series method. The general procedure for solving such problems was discussed, including the need to extend the series to include values of n less than 0 in order to avoid getting all coefficients as zero. The use of recursion and computing limits was also mentioned as a way to solve for the coefficients of x^n.
Schfra

## Homework Statement

I need to solve the DE

y’ = x^2y

using the power series method

## Homework Equations

y = sum(0->inf)Cnx^n
y’ = sum(1->inf)nCnx^(n-1)

## The Attempt at a Solution

I plug in the previous two equations into the DE. What is the general procedure for these problems after that?

I believe I can bring the x^2 into the summation to get

sum(1->inf)nCnx^(n-1) - sum(0->inf)Cnx^(n+2) = 0

I’m not sure what to do after this.

Now you need to solve for the coefficient of ##x^n## for each value of ##n##. So for example ##2c_2 - c_{-1} = 0##. It looks to me like you need to extend your series to include values of ##n## less than 0. Otherwise, all of your coefficients will be zero.

tnich said:
It looks to me like you need to extend your series to include values of ##n## less than 0. Otherwise, all of your coefficients will be zero.
Can you explain what you mean by this and why I have to do this? I did get 0 for my coefficients while attempting to solve earlier.

Schfra said:
Can you explain what you mean by this and why I have to do this? I did get 0 for my coefficients while attempting to solve earlier.
There are a couple of tricks here. First, assume that ##nc_n - c_{n-3} = 0## for all ##n \in \mathbb Z##. You end up with a recursion. Actually you end of with three recursions. For two of them you easily compute a limit and dispense with them. The other one gives you ##0c_0 - c_{-3} = 0##. What does that imply ##c_{-3n}##? About ##c_0##?

tnich said:
Now you need to solve for the coefficient of ##x^n## for each value of ##n##. So for example ##2c_2 - c_{-1} = 0##. It looks to me like you need to extend your series to include values of ##n## less than 0. Otherwise, all of your coefficients will be zero.
I disagree. The solution to the differential equation has a smooth behaviour around ##x = 0## so expanding it in a Maclaurin series should be perfectly fine.

Also note that you have no ##c_{-1}## so clearly ##c_2 = 0## (which agrees with the exact solution).

## 1. What is the power series method for solving differential equations?

The power series method is a technique used to find solutions to certain types of differential equations. It involves representing the solution as a series of powers of a variable, typically x. By plugging this series into the differential equation and solving for the coefficients, a general solution can be obtained.

## 2. When is the power series method applicable for solving differential equations?

The power series method is applicable for solving linear differential equations with variable coefficients, including ordinary and partial differential equations. It is particularly useful for finding solutions near a specific point, such as an initial condition.

## 3. What are the advantages of using the power series method for solving differential equations?

One advantage of the power series method is that it can handle more complex and non-standard differential equations that may not have a known analytical solution. It also allows for the determination of a general solution instead of just a specific solution.

## 4. What are the limitations of the power series method for solving differential equations?

The power series method may not work for all types of differential equations, particularly those with singularities or discontinuities. It also requires a significant amount of algebraic manipulation and may not always yield a closed-form solution.

## 5. Can the power series method be used for solving nonlinear differential equations?

Yes, the power series method can be used for solving some types of nonlinear differential equations. However, it may require additional techniques, such as substitution or iteration, to handle the nonlinearity.

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