Higher Order Linear Differential Equations with Constant Coefficients
All the numbers we know are called real numbers and can be set on a straight line called the line of the real numbers. So numbers like 1, 2, -7, 4.78, -3.14, 2/5, the square root of positive numbers, are real numbers. They are set on the real numbers straight line and we can apply to them all the operations, like addition, multiplication, etc.
What if we want to calculate the square root of a negative number? There are no real numbers that could be the solution of the square root of a negative number. In this case the imaginary numbers are used.
Consider the following two complex numbers:
We can apply to them the following operations, as shown:
The complex numbers must be in Cartesian form. We add the real parts together and the imaginary parts together.
Phasors (phase vectors) is a very useful mathematical tool that converts a differential equation, with the time as the independent parameter, into algebraic form. Then the equation can be solved much easier and the solution is finally converted back to its initial form.
A phasor is actually a complex number that represents the amplitude and the phase of a sinusoid, with its time variable neglected. The amplitude and the angular velocity must be time invariant. Since the factor ωt is known, it can then be neglected. Two sinusoids and their phasors follow:
To solve equations systems in algebraic way, we have to end up to an equation that has only one unknown variable. To do this, we can add two equations together to zero out unknown variables or we can substitute a variable from one equation to another.
This technique will be explained by the following example: