The basic concepts ordinary and partial differential equations (ODEs, PDEs), order, general and particular solutions, initial value problems.

If the derivatives in the equation are ordinary, the equation is said to be a normal differential equation.

Example of ordinary differential equation:

y′+y=0

If the derivatives in the equation refer to two or more independent variables, the equation is said to be a partial differential equation.

Example of partial differential equation:

xd2ydx2+(2x−1)dydx+xy=0

If the nth derivatives are the largest derivative in the equation, that equation is said to be of order n.

xd2ydx2+(2x−1)dydx+xy=0

From the above example, it can be observed that the order of differential equation is 2 because the largest order of differential equation is 2.

The general solution of the ordinary differential equation is the solution in which the number of arbitrary constants equals the order of the differential equation.

Suppose that y=x2+2x+c differential equation, then the solution of ordinary differential equation is y=(c1+c2x+c3x2)e−x.

Initial value problem is a differential equation with given initial condition is 3y″+2y′=0, with initial value y′(0)=−3.5,y(0)=3.

Partial solution is the solution obtained with the problem of the initial value, which is used to find the values of the integrated constants.