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

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.