Solving a third order DE with Maple

[ehilge]

Homework Statement

This problem relates to using a numerical analysis program called Maple. My question is not related to how to run the program, but the appropriate steps needed to solve the equation. So even if you aren't familiar with the program, please try to help me out.

Consider the third order non-linear differential equation: (y'')(y''')=y

a) Solve this equation in Maple.

b) Find all polynomial soltions y(x)= a₀ + a₁x + a₂x² + ... a₅x⁵ of this equation (It turns out that these are all solutions that are polynomial)

Homework Equations

none that I'm aware of at this point

The Attempt at a Solution

a) the output from Maple is shown in the attached screenshot

b) I have thought over this for awhile and I can't think of a good methods to finding the polynomials. The only ways I have been taught to solve a high order DE is through reduction of order, which is impossible since I don't have a solution I can work with. And variation of parameters, which I can't see working very well. Also, I'm thinking that the solution to (b) might have something to do with interpreting the solution found in (a) since there is a polynomial part in there. But again, I don't see how to create a polynomial solution out of it. So, any ideas on how to find the polynomial solutions, and particularly what steps I could take with the computer to get there?
thanks!

 

[mighty2000]

Thank you for your question. I am not familiar with the Maple program, but I can offer some general steps that may help you solve this equation.

First, let's rewrite the equation as follows:

y''' = (y/y'')'

Now, let's try to find a solution by assuming that y is a polynomial of the form y(x) = a0 + a1x + a2x^2 + ... + anxn. Substituting this into the equation, we get:

y''' = 6anxn-3 + 12an-1xn-2 + 6an-2xn-1

(y/y'')' = (3anxn-2 + 2an-1xn-1 + an-2xn)/(2anxn-2 + 2an-1xn-1)

Setting these two expressions equal to each other and simplifying, we get:

3anxn-2 + 2an-1xn-1 + an-2xn = 6anxn-3 + 12an-1xn-2 + 6an-2xn-1

Rearranging and grouping like terms, we get:

(3a - 6)anxn-2 + (2a - 12)an-1xn-1 + (a - 6)an-2xn = 0

This is a system of equations in terms of the coefficients a0, a1, a2, ... , an. Solving this system will give us the polynomial solutions to the original equation.

I hope this helps. Let me know if you need further assistance.