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 1st order ODE nonlinear

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

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

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

The most general nth 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