**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**