3rd Order Linear Homogenous DE from Solutions

[SArnab]

Homework Statement

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

e^$\pi$t, te^$\pi$t 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^$\pi$t:
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^$\pi$t, te^$\pi$t 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^$\pi$t:
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$\pi$ + $\pi$²)(r+1)

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

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

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

Therefore the DE would be:

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

Thanks for the help!