Differential Equations: Cracking 1st order and degree

-
Amazing Maze Garden / Enjoy a Walk in the Garden Maze

Mazes are a very accurate representation of the process that goes behind solving differential equations. When initially experimenting with them, without any prior knowledge, they may seem impossible. However, when provided with a bird's eye view of the layout, the process suddenly seems simple. And with practice, it becomes easier and quicker to reach the final destination. This description is most applicable for first order, linear differential equations, a set of differential equations of the simplest kind.


How do you solve 1st order, linear differential equations? 

A first order, linear differential equation has an order and degree of 1 [refer to previous posts for clarification]. It can also be represented in the following form:

𝑑𝑦/𝑑𝑥+𝑝(𝑥)𝑦=𝑞(𝑥)

In order to solve this set of differential equations, a particular series of steps must be followed:
  1. Rearrange the differential equation to the form shown above
  2. Substitute 𝑦 as a product of 2 functions called 𝑢 and 𝑣, altering 𝑑𝑦/𝑑𝑥 as well
  3. Factor out the parts involving 𝑣 and equate this new expression to 0
  4. Use separation of variables and integration to find 𝑢
  5. Substitute this value into the equation of step 2
  6. Use that to find the 𝑣, and then use 𝑢 and 𝑣 to find the final solution 𝑦
Obviously, this process sounds severely vague and complicated. Hence, this process will be applied to an easy example below: [If the slides appear to be unclear, refer to the presentation using the link provided at the end of this post]






The case above was a classic first order linear differential equation. Each step is crucial in obtaining the final result and even a tiny error can have huge repercussions. This method is fairly complicated, yet it is special since it works for the entire set of first order, linear differential equations. In general, differential equations do not have such a direct, elegant method. Learn more about differential equations through this oversimplified presentation.

Comments

Post a Comment

Popular posts from this blog

Examining properties of flight

-
Image
Motion in air is a prime example of how  physics and mathematics dominates our lives, exemplifying their synergy. Most physics laws and calculations need complex mathematics to be proven/justified, but mathematics by itself isn't particularly useful in terms of its application in the real world. Thus, this page would mainly show how in-flight motion employs the need to use both physics and mathematics together for better understanding.
Read more

Differential Equations: Rising complexity

-
Image
After careful examination of the last post, it may seem easy to master first order, linear differential equations . However, the previous example was only the tip of the iceberg, missing important elements and possible functions that could've turned that seemingly simple problem into a waking nightmare. This post will demonstrate the difference using a slightly more challenging example . It is strongly advised to refer to the previous post for context of the content in this post.
Read more

Differential Equations: Classification

-
Image
Differential equations exhibit fascinating variation , with a division as diverse as regular, algebraic equations. Possessing knowledge about the various types and correctly applying that knowledge to identify a particular one is a key step in solving any differential equation. This is because, it gives an idea/hint about which method can be used in that particular scenario, saving time and energy. While the classification can get very complex, most of the basic, yet crucial, differences are highlighted here.
Read more