What is a differential equation?
An equation that involves one or more derivatives of an unknown function is called a differential equation. The order of the highest derivative included in a differential equation defines the order of this equation.
Examples

y ' = 3x ,
the order of the highest derivative is 1 (y ' ) so the order of this differential equation is 1.

y '' + y' + y = 3x ,
the order of the highest derivative is 2 (y '' ) so the order of this differential equation is 2.

2 y ''' + y'' + y^{ 4} = 3x ,
the order of the highest derivative is 3 (y ''' ) so the order of this differential equation is 3.
y = f(x) is a solution of a differential equation if the equation is satisfied upon substitution of y and its derivatives into the differential equation.
Example:
Verify that y = C*e^{ 4x} + e^{ 3x}, where c is a constant, is a solution to the differential equation
y '  4y = e^{ 3x}
y ' is given by
y ' = 4C*e^{ 4x} + 3e^{ 3x}
We now substitute y ' and y into the left side of the equation and simplify
y '  4y = 4C*e^{ 4x} + 3e^{ 3x}  4 (C*e^{ 4x} + e^{ 3x})
= 4C*e^{ 4x} + 3e^{ 3x}  4C*e^{ 4x}  4e^{ 3x}
= 4C*e^{ 4x}  4C*e^{ 4x} + e^{ 3x} (3  4)
=  e^{ 3x}
Which is equal to the left side of the given equation and therefore y = C*e^{ 4x} + e^{ 3x} is a solution to the differential equation y '  4y = e^{ 3x}.
Most of the work on differential equations consists in solving these equations. For example to solve the following differential example
y ' = 2x
Let us integrate both sides of the given equation as follows
ò y ' dx = ò 2x dx
which gives
y + C1 = x^{ 2} + C2
where C1 and C2 are constants of integration. The solution y of the above equation is given by: y = x^{ 2} + C, where C = C2  C1.
More references on
Differential Equations
Differential Equations  Runge Kutta Method
