Differential Equations

Differential Equations
A differential equation is an equation that involves one or more derivatives. Differential equations are classified by:
1. type (ordinary or partial)
2. order (that of the highest order derivative that occurs in the equation)
3. degree (the exponent of the highest order derivative)
For example :

is an ordinary differential equation, of order one and degree one .
The differential equation is an ordinary differential equation, of order three and degree one , the differential equation

is an ordinary differential equation, of order three and degree two. But the differential equation


is a partial differential equation, of order two and degree one.
Solution of differential equation
A function is said to be a solution of differential equation if the latter is satisfied when and its derivatives are replaced throughout by and its corresponding derivatives.
For example, if are any constants, then

is a solution of the differential equation


General solution and particular solution
A particular solution of a differential equation is any solution that is obtained by assigning specific values to the arbitrary constants in the general solution.
Geometrically, the general solution of a differential equation represents a family of curves known as solution curves. For instance, the general solution of the differential equation is (General solution)

Figure 1 shows several solution curves corresponding to different values of
Particular solutions of a differential equation are obtained from initial conditions
placed on the unknown function and its derivatives. For instance, in Figure 1, suppose
you want to find the particular solution whose graph passes through the point
This initial condition can be written as (Initial condition)
Substituting these values into the general solution produces which implies
that So, the particular solution is (Particular solution)
Example 1:
Verify that is a solution for the differential equation
Then find the particular solution determined by the initial condition
Solution :

Then is a solution for the differential equation
Furthermore, the initial condition yields

and you can conclude that the particular solution is :