so now let us define that the powers of the Earth are the worst thing to ever do to this world. It is not that what I am thinking of bears any relavence to what it out there in the whole world wide universe, but there is something out there that is meant to be seen in this whole for the entire human race. Now such that anything would do, especially that anything there is to is possible. And so I end this brief essay for doing just that where everything is reserved and now there are losers and there are winner. I just happen to be a winner. The end.

Continuing to post because it's so old.

## Hmm...

## Tuesday, November 21, 2006

## Wednesday, November 08, 2006

## Friday, November 03, 2006

### MATH307 Ordinary Differential Equations | 060828

**Differential Equation** **(DE)** | any equation involving the derivatives of dependent variables with respect to one or more independent variables

If there is only one independent variable then we have an **ordinary differential equation (ODE)**. Otherwise a **partial differential equation (PDE)**.

**Order of a DE** **|** the order of the highest derivative it contains

A DE is said to be **linear** if the dependent variable and its derivatives occur **only linearly** (to the first power), otherwise it is **non-linear**.

(y^i)^2+2sin(xy)=0 1^{st} order ODE nonlinear

3y^iii+2y^i+y=e^x 3^{rd} order ODE linear

3y^iii+2y^i+y=e^y 3^{rd} order ODE nonlinear

d2u/dx2+2d2u/dxdy+du/dx=x^2+y^2 2^{nd} order PDE linear

The most general n^{th} order ODE takes the form

G[x,y,y^i, y^ii, … , y^(n)] = 0 (1)

or in principle we can write as

g^(n)=F(x,y,y^i,…,y^(n)) (2)

Any function y=f(x) that satisfies (1) [or (2)] identically in some interval I is called an **explicit function** of (1) [or (2)] in I.

An equation g(x,y)=0 that can be solved to give one or more explicit solutions is called an **implicit solution**.

Show that y=e^2x + 2e^x is an explicit solution of y^ii-3y^i+2y=0

y=e^2x + 2e^x, y^i = 2e^2x+2e^x, y^ii = 4e^2x+2e^x

y^ii-3y^i+2y=4e^2x+2e^x-3

= 0e^2x+0e^x=0 **OK**