ex-xian

I've loved and used Euler's equation for awhile now, that is

e^(pi*i)+1=0,

but I've never set out to prove it for myself. Well I did today, but I'm stuck. Here's where I went. If I can show that e^(ix) = cos x + i sin x, then when x=pi, it works out.

Let z=cosx + isinx
dz/dx = -sinx +i cosx
= (i)^2 sinx + i cosx
= i (cos x + i sin x)
= zi.

I don't know what to do next. It looks like I can divide by z, and

1/z * dz/dx = i.

If I can integrate the left side I'd get (I think) ln z = xi + a constant. Which would work out nicely, I think. But don't I have to integrate the right side? What does it mean when you have a function times this.....

...wait a minute.....in the middle of writing this, I realized that this:
dz/dx = zi, is a first order diffy q. I've not had that class formally..just what I've picked up in quantum physics.

So the solution to the DE is z=e^(ix)? I just finished the proof while I was typing this...cool...oh well, I'll go ahead and post just in case I made a mistake.

Shadowy Man

It always seemed to me that the way to solve differential equations was to guess at a solution and then show that the solution works.

Not very satisfying, but then again I'm not a mathematician, so it was just fine with me.

ex-xian

Funny...that is almost verbatim what my physics prof told me.
But she moved to CA.

Undercurrent

I've always taken it from the power series. By definition:

exp(z) = 1 + z/1! + z^2/2! + z^3/3! + ...

letting z = ix, and using i^2 = -1, i^3 = -i, i^4 = 1, &c, we get:

exp(ix) = 1 + ix/1! - x^2/2! - ix^3/3! + x^4/4! + ix^5/5! - ...

Separating out the terms with i in them, we get:

exp(ix) = (1 - x^2/2! + x^4/4! - ...) + i(x - x^3/3! + x^5/5! - ...)

The first bracketed expression is the power series for cos(x) and the second is the power series for sin(x), therefore:

exp(ix) = cos(x) + i*sin(x)

QED.

ex-xian

That's clever. I wouldn't have thought of that.

Undercurrent

Just a point here. The complete solution to the DE is z = k*exp(ix), where k is a constant. What you prove through the DE is that cos(x) + i*sin(x) = k*exp(ix) for some k. To show that k is, in fact, equal to 1, you need to aftewards plug in a value (x = 0 is convenient) for both sides and solve.

lpetrich

A trick for improving one's "guessing" is the method of undetermined coefficients. You guess that a solution has a certain form, aside from the values of certain coefficients, plug that form in, and then find the coefficient values that give a solution (if any).

Thus, for dz/dx = i*z

one guesses that this is some exponential function z = a*exp(b*x)

Thus,

dz/dx - i*z = a*b*exp(b*x) - a*i*exp(b*z) = a*(b - i) * exp(b*z)

with solutions:

a undetermined (constant of integration)
b = i

For differential equations, one will get constants of integration; these can be fixed by the problems' boundary conditions. In fact, one can impose the satisfaction of boundary conditions on one's guessed solutions.

Thus, in our current problem, z(0) = 1, which implies that b = 1

lpetrich

"Guessing" may seem like one does not know what one's doing, but certain sorts of equations always have certain forms of solutions, which helps considerably.

A common example is homogeneous linear differential equations, like the one in this problem. Their solutions are always linear combinations of exponential functions of linear combinations of the independent variables.

ex-xian

Do you mean a = 1?

Anyway, thanks for the tips on solving the DE's. As I indicated, I've not had a formal class in that subject. But I will have after this semester. I'll have to remember this so I can look all cool and stuff by already knowing how to do it.

ex-xian

But they can also be written in terms of trig functions, right? Do you just choose whichever one is simplest?

Also, is there a similiar "method" for solving, I'm not sure if this is the correct terminology, second order DE? For example, depending on the energy, schroedinger's equation (with constant potential and omiting constants) becomes:
d^2y/dx^2 = E
or
-d^2y/dx^2 = E, where y=wave function.
I know that there are different solutions for each, but is there any way to solve them when they are more complicated.

And what about partial DE?