Undetermined Coefficient

The Undetermined Coefficient is one of the ways to find the particular solution to a non-homogeneous differential equation.

The advantage with this approach is reduces the problem to an algebra problem. However, this may work only on limited classes of g(t).

We can write y(x) as,

 Where,

Let's solve this given SODE,

 We first solve for yh,

 Since they are real roots, we have yh as,

For yp, we have to guess what it looks like based on g(x). The table below will help us to determine that.

Since g(x) = 3e^(5x), then our yp guess is,

We can plug yp guess to get our actual yp,

we now have y(x) and y'(x) as follows,

To get c1 and c2, we set x = 0 with the above equations,

Now, we have our actual solution,

Separable First-Order of Differential Equation

A first-order differential equation is said to be separable if , after solving for the the derivative,

 the right-hand side can be factored by function of x and function of y,

If the factoring is not possible, this means that the derivative is not separable.

Let's have an example,

First, we check if it is separable,

Since it is separable, we can continue solving it by integrating both sides:

Another application for this is with Lotka-Voltera model of predator y - prey x dynamics.
It is given by,

 Where,

 Independently, they can't be solve in linear method. However, the dependence of the predator prey population can be solve by,

 This now becomes a separable fode as proof below,

After we integrate both sides, we are left with an implicit function.

Fourier Series Approximation

Fourier Transformation is a function f(t) derived from a given function and representing it by a series of sinusoidal functions. It is expressed as:

where,

It is used in multiple fields like signal and noise estimation, seismographs, and filtering.

To show how it works, we'll give the Fourier series of the given functions:

Since both of these functions is at t(0), we need to shift the charging formula.

If plotted, it will result to this:

Note that which equation Q(t) to use depends on time t,

To solve a0,

To solve an,

To solve bn,

By plugging the values, we will have f(t) as,

As we increase the approximation term n , the series approximation improves as well.

LU Factorization

LU Factorization is used to factor matrix A as the product of lower-triangular matrix L and upper-triangular matrix U.

A = LU

In order to do such decomposition, one must have the knowledge of Gaussian Elimination. 

To better understand how this works, given the matrix M below, we'll get it's LU Decomposition.

To get U we will do the following transformation: 

A lot of us (or maybe just me) have at least once imagined what it be like to become spy. With their high-tech gadgets, jaw-dropping acrobatic stunts, and, more importantly, a covert life. We often see in movies that they pass around unreadable messages that seems to make no sense. But we eventually see that every message is just puzzle that is yet to be decoded.

Let's have an activity where we will try to decode this message:

The given encoded message was derived from this equation:

In order to get the original message (M),  we can rewrite the equation to:

And we already have an agreement with the sender that our encoding matrix will be:

We need to first solve for the inverse of C. This can be done via LU Decomposition.

Now, it is just a matter of plugging in our values:

The result is represented by their ASCII values, so we can convert each of them.

And we will have the original message,

"Sherman - Morrison Formula!"

Which is actually the name of the formula we used to encode our message!

A VERY BUSY CHRISTMAS BREAK!

I know you already heard many times that being student is hard and this post will only just whine more about the stressful life of a student because this blog post is made by a stressed-out BSIT 3rd year student - ME. 

Third year is the time of most bachelor degrees where students are flooded with demanding, time-consuming, soul-eating, sleep-repellant, dietetic, and stressful major subjects. And I am currently experiencing this dilemma. We were given so many things to do - BUT I can’t say we were given insufficient time. It’s just that we can’t really work on our tasks everyday because it’s the holiday season – it’s CHRISTMAS TIME for crying out loud.  We also need to celebrate with our families.

We were given the following tasks:

Software Engineering

·        System Design Description

·        System Test Plan

·        At least 20% of the system properly running

Technoprenuership 2

·        Initial drafts of all chapters of the business plan

Future Technology

·        Technology Proposal

Network Technology

·        Powerpoint Presentation for the Topic Sharing

Multimedia

·        Design a logo for our college

·        Study 166 videos for the exam

Project Management

·        Powerpoint Presentation for the Topic Sharing

Thinking only of the list of tasks to do already stresses me out. I already finished my logo and half of the SE tasks are almost done but the others are still to be worked. It is really hard especially some group tasks needs to be managed by one person and some of those are managed by me so I have to allot time to plan the flow on how we will finish the task and the proper dissemination of tasks to the others.  Another bad news is we are to return immediately in January 2. That means I (and my group mates) only have five days to finish the remaining work.

I know some of you might think, “So why did you even make this post if you were busy?” or “Stop whining and do your work!” Aside from being a student, I’m also a teenager who gets easily distracted with other things like facebook, 9gag, animes, mangas, and other things so I rather blog about this than browse 9gag or read some updates in facebook(This is also my chance of updating my blog :P ). But no worries, I have been in the same situation many times before and I have finished all my tasks. I know that this kind of pressure may come back when I work in the future so it’s better that as of now – as a student – I can calmly manage my time and work.

So that’s it, WISH ME LUCK! 

[NOTE: I'm doing this 3 in the morning so I wanna say sorry ahead of time for the unchecked typographical and grammatical errors.]

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