# Differential Equations: Second Order Equations

peace-Econ

## Homework Statement

Find a second order differential ewuation for which three functions y=2e^-t, y=2te^-t, y=e^(-t+1) are solutions.

## The Attempt at a Solution

Homework Helper
Gold Member

Can you solve this de:

y'' - 4y' + 4y = e2t

and if so, what method would you use?

Hint: This isn't an idle question.

Homework Helper

## Homework Statement

Find a second order differential ewuation for which three functions y=2e^-t, y=2te^-t, y=e^(-t+1) are solutions.

## The Attempt at a Solution

If those functions were independent, this would be impossible but $e^{-t+ 1}= e^{-t}e^1= e e^{-t}$, a constant times $e^{-t}$ so you really have only two independent solutions.

Do you know what the "characteristic equation" of a linear differential with constant coefficients is? Such an equation will have $e^{ax}$ and $e^{bx}$ as independent solutions if and only if its characteristic equation is $(r- a)(r- b)= 0$.

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