3rd Order Linear Homogenous DE from Solutions

[SArnab]

Homework Statement

Find a third order, linear, homogeneous DE which has the following solutions:

e^\pit, te^\pit and e^-t

Homework Equations

Standard form of a third-order linear homogenous ODE:

Ay^''' + By^'' + Cy^' + Dy = 0

The Attempt at a Solution

I tried deriving the characteristic equation given the r values of the solutions.

For e^\pit:
r = \pi, therefore one part of the equation is (r - \pi)

For e^-t:
r = -1, therefore another part of the equation is (r+1)

But I can't figure out what to do for the second one.

 

[LCKurtz]

Homework Statement

Find a third order, linear, homogeneous DE which has the following solutions:

e^\pit, te^\pit and e^-t

Homework Equations

Standard form of a third-order linear homogenous ODE:

Ay^''' + By^'' + Cy^' + Dy = 0

The Attempt at a Solution

I tried deriving the characteristic equation given the r values of the solutions.

For e^\pit:
r = \pi, therefore one part of the equation is (r - \pi)

For e^-t:
r = -1, therefore another part of the equation is (r+1)

But I can't figure out what to do for the second one.

Remember that a repeated root $r$ of the characteristic equation gives rise to the solution pair $\{e^{rt},te^{rt}\}$.

 

[SArnab]

Ah, so it just has a repeated root.

So the characteristic equation would be:

(r-\pi)²(r+1)

Factor that out and we get:
(r² - 2r\pi + \pi²)(r+1)

r³ - 2r²\pi + r\pi² + r² - 2r\pi + \pi²

r³ - 2r²\pi + r² + r\pi² - 2r\pi + \pi²

r³ + r²(-2\pi + 1) + r(\pi² - 2\pi) + \pi²

Therefore the DE would be:

y^''' + (1-2\pi)y^'' + (\pi² - 2\pi)y^' + \pi²y = 0

Thanks for the help!