Introduction to Differential Equation

Differential Equations

1. Type

Ordinary Differential Equations

ex ) y" + y' = 0 , y" + 7 x( y' ) ^3 = 3x , ...

Partial Differential Equations

ex )

, ...

2. Linearity

Linear Differential Equations 

ex) y" + xy' = 7x,  8y"' + 3y' = xy, ...

Nonlinear Differential Equations

ex) (y"")^19 + (y")^ 30 - xy = 30x, ...

3. Order

1st, 2nd, 3rd, ... , n-th Order Differential Equations

ex)

y' + y^3 = 0 -> 1st order differential equation

(x^50)y" + (y')^300 + y = x -> 2nd order differential equation

x(y"')^20 + y = 3x^398 -> 3rd order differential equation

Linearity

Definition of linearity

Function L has additivity and Homogeneity.

Let's find whether some functions have linearity.

In these example, some functions don't have linearity,

f(x) = ax, a is in real number, has linearity.

All linear functions don't have linearity.

f(x) = ax + b, b is not 0, is a linear function but it doesn't have linearity.

If functions are linearity, we can analyze and expect easily.

Let's see the graph below.

In this graph, If we want to find quantity,

 we don't have to estimate quantity for length of time of 287.

We just estimate quantity for length of time of 142 and 145.

Because we can use additivity in the graph above.

f( 142+145 ) = f( 142 ) + f( 145 )

Another method, We just estimate quantity for only length of time of 1.

Because we can also use homogeneity in the graph above.

f( 287 ) = 287 f( 1 )

If a graph doesn't have linearity, we can't expect quantity easily.

Look at this graph, 

we are unable to easily assume a function of the graph,

so, we can't easily expect quantity of length of time of 287

before that time.

Cramer's Rule

Before you see this post,

see

Properties of Determinant

I've used Gauss-Jordan Elimination 

to solve simultaneous equations.

There is another method to solve simultaneous equations,

Cramer's Rule

You can understand how to solve simultaneous equations by Cramer's rule.

Zeckendorf's theorem

Before you see this post,

see

Fibonacci Number

Zeckendorf's theorem states 

that every positive integer can be represented uniquely 

as the sum of one or more distinct Fibonacci numbers 

in such a way 

that the sum does not include any two consecutive Fibonacci numbers.

Vector in Coordinate System

Before you see this post,

see

Eigenvalue in Matrix

Diagonalizable

Total differential

Unit Vector

pic. 1

Let formulas be supposed like that pic. 1.

pic. 2

We can express the vector v like pic. 2.

pic. 3

We can find relationship 

between total differentials in Cartesian and general coordinate system

in pic. 2 and pic. 3.