Finding Solutions to 4th Order Differential Equations

[Mathgician]

Homework Statement

solve the given differential equation:

xy^(4) + 6y'" = 0

Homework Equations

No idea, I need something to get me started

The Attempt at a Solution

auxiliary equation:

m^4 + 6m^3 = 0

m^3(m+6)=0
3 roots with m = 0 and the other root is m = -6

so I got y = C1 + C2lnx + C3(lnx)^2 + C4x^-6

but, it is not the solution. Your help is greatly appreciated. Thank You.

 

[cristo]

I don't quite know what you're doing here. The method you are using here seems to be the one that you use for a second order equation, however this is an equation of 4th order.

Take a look at the equation and try and spot a solution. In particular, you have a fourth derivative multiplied by x, and a third order derivative. Now, you know a function whose fourth derivative multiplied by x is equal to its third derivative (modulo constants). The function is xⁿ. So, try the general solution y=xⁿ. Plugging this into the ODE will give you a condition on n, which will enable you to obtain the particular solution.

 

[HallsofIvy]

Your auxiliary equation is wrong. You appear to have used the auxiliary equation for a "constant coefficients" equation when you have a variable coefficient- and then used the "Euler type" solution.

One way to handle this is to multiply then entire equation by x³ so that you get x⁴y^(iv)+ 6x³y'''= 0, an "Euler-type" or "equipotential" equation. If you let y= x^r you will get the correct auxilliary equation for this d.e.

Much simpler is to let u= y'''. Now the equation is xu'+ 6u= 0, a simple, separable first order differential equation. After you have solved that, let y'''= u= your solution and integrate three more times.

 

[Mathgician]

so the auxiliary method only works for 2nd order? Can you guys give me little more insight or any websites that would explain it to me further because my books was very vague in explain my problem or there were no example of such problem in the book. I hate my textbook. Thank you for your insights guys :shy:

 

[Mathgician]

can you give me hint on how to solve an equation that involves a particular solution? Thank you