Friday, November 30, 2018

Differential Equations

Differential Equation (DE) is one of the courses I have studied. I want to share notes on this blog about the material of Differential Equations. For those of you who want to follow these notes, please look at the Differential Equations category.

1. Definition of Differential Equations
Differential Equation is an equation that contains an unknown function and its derivatives. If in the equation, there is only one independent variable involved then it is called ordinary differential equation (ODE) and if more than one independent variable is involved it is called partial differential equation.

2. Forming Differential Equations
If a function is known then to form a differential equation, it is derived to the order (level) to the number of constants contained in the function and then eliminate constants based on the number of constants + 1 equation.

For example, given the function $ y = A sin \ 3x + B cos \ 3x $. We will form the differential equation of the function.
First, we lower y to x to the second derivative because there are two constants we want to remove that are contained in the function, namely A and B.

$y=A sin \ 3x + B cos \ 3x \ .... (1)$
$\frac{dy}{dx} = 3A cos \ 3x - 3B sin \ 3x \ .... (2)$
$\frac{d^2y}{(dx)^2} = -9A sin \ 3x - 9B cos \ 3x \ .... (3)$

Second, we eliminate the constants A and B by using press 1 and 3, so we get:
$ \frac{d^2y}{(dx)^2} + 9y = 0 $
So, the cluster differential equation of the curve is $ \frac{d^2y}{(dx)^2} + 9y = 0 $ or it can also be written as $ y"+ 9y = 0$

3. Complete Differential Equations
Resolving differential equations is finding y=f(x) that satisfies a PD and this is what is called a PD solution.
a. General Solution: A solution that is stated explicitly or implicitly that contains all possible solutions to a domain. This general solution contains n arbitrary constants.
b. Special Solutions: Solutions that do not contain any arbitrary constants.
c. Singular Solution: In some cases there are other solutions to the equation given by the solution that cannot be obtained by giving certain values ​​to any constants of the general solution.

Thus this brief discussion, hopefully can be understood.

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