# Thread: Determine a Dif EQ

1. ## Determine a Dif EQ

For each of the following, determine a differential equation (or a system of Dif EQ's), which has the expression of what is below for the general sol'n.

1.) $\displaystyle y=c_1\cos(\sqrt{2}t)+c_2\sin(\sqrt{2}t)$

2.) $\displaystyle y=c_1e^{4t}+c_2te^{4t}$

3.) $\displaystyle \textbf{X}=c_1\left(\begin{array}{r}0\\1 \end{array}\right)e^{-3t}+c_2\left(\begin{array}{r} 1\\0 \end{array}\right)e^{2t}$

2. Let's think about characteristic equations:

1) Imaginary Solutions?

2) Real, Repeated Solutions?

3) Real, Unequal solutions?

3. Originally Posted by Auxiliary
For each of the following, determine a differential equation (or a system of Dif EQ's), which has the expression of what is below for the general sol'n.

1.) $\displaystyle y=c_1\cos(\sqrt{2}t)+c_2\sin(\sqrt{2}t)$
$\displaystyle \frac{d^2y}{dt^2} + 2y = 0$

Originally Posted by Auxiliary
2.) $\displaystyle y=c_1e^{4t}+c_2te^{4t}$
$\displaystyle \frac{d^2y}{dt^2} - 8\frac{dy}{dt} + 16y = 0$

4. Originally Posted by kalagota
$\displaystyle \frac{d^2y}{dt^2} + 2y = 0$

$\displaystyle \frac{d^2y}{dt^2} - 8\frac{dy}{dt} + 16y = 0$
Thanks kalagota! How exactly did you come up with those by the way? Was it a trial and error type of deal?

5. Originally Posted by Auxiliary
Thanks kalagota! How exactly did you come up with those by the way? Was it a trial and error type of deal?
Originally Posted by kalagota
$\displaystyle \frac{d^2y}{dt^2} + 2y = 0$
Let's turn this around for a moment:

How would you go about solving
$\displaystyle \frac{d^2y}{dt^2} + 2y = 0$