Question concerning basic Diff.Eq. 2nd order trick

[astropi]

question concerning basic Diff.Eq. 2nd order "trick"

Homework Statement

First of all, this is NOT for homework. In fact, this is a conceptual question. Now then, let me address my problem by presenting a simple example. Let's say you have a second order equation with initial conditions:

y(-1) = 2 & y'(-1) = 1

You can solve the D.E. by just applying the boundary conditions as usual, but I also understand there is a "trick". The trick is to rewrite the general solution. In other words, you can solve it the normal way by applying your ICs on (note the actual equation does not matter):

y = c_1e^{r_1t} + c_2e^{r_2t}

or you can use this "trick"

y = d_1e^{r_1(t+1)} + d_2e^{r_2(t+1)}honestly, I don't remember where this trick comes from. It's supposed to simplify the equation, because then when you apply -1 to t your exponents go to 0. However, could someone explain WHY this is valid? When exactly can you use this, and when can you not? Thanks, hopefully I've been clear about this.

-astropi

ps I tried to use [tex] and the thing totally messed up! What's up with that?

 

[HallsofIvy]

$$e^{r_1(t+1)}= e^{r_1t+ r_1}= e^{r_1t+ r_1}= \left(e^{r_1}\right)\left(e^{r_1t}\right)$$

And the constant $e^{r_1}$ can be incorporated into the constant multiplying $e^{r_1(t+1)}$.

That is, $c_1= d_1e^{r_1}$.

You tried to edit and the LaTex messed up? That's an annoying bug on this board. The LaTex will come out right the first time but if you try to edit it you can get some really weird things! Just click on the "refresh" button on our internet reader and it should correct. I tend to automatically click on "refresh" any time I edit LaTex.

 

[astropi]

$$e^{r_1(t+1)}= e^{r_1t+ r_1}= e^{r_1t+ r_1}= \left(e^{r_1}\right)\left(e^{r_1t}\right)$$

And the constant $e^{r_1}$ can be incorporated into the constant multiplying $e^{r_1(t+1)}$.

That is, $c_1= d_1e^{r_1}$.

You tried to edit and the LaTex messed up? That's an annoying bug on this board. The LaTex will come out right the first time but if you try to edit it you can get some really weird things! Just click on the "refresh" button on our internet reader and it should correct. I tend to automatically click on "refresh" any time I edit LaTex.

Thanks for the reply, and the note on the latex bug!
Now I do understand that you multiply by an extra $$e^{r_1}$$, but just to clarify, you actually multiple both sides of the equation by that value (as you must in order to conserve the original equation), but that extra value on the left-hand side is just incorporated into the constant. Is that correct? Thanks in advance.