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