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
