Marvelous Mathematics and Fantastic Physics

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.

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.

Verification of a differential equation is exactly like plugging in the last piece of a jigsaw puzzle; the solution is inserted back into the differential equation to ensure that it qualifies its conditions . Apart from the satisfaction of finding out a successful solution that may have taken hours of effort, learning to verify accurately is also important for various other reasons. Fortunately, verification is far simpler than actually solving most differential equations.

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.

Differential Equations hold a very special place in the field of mathematics, forming the strong interlocking chain that bridges the gap between abstract mathematics and each of the practical sciences . Principally, differential equations are a subset of calculus, as every differential equation works with functions and their derivatives . Much like algebraic equations, some differential equations are easy enough to be solved by any novice, while others have remain unsolved and teased centuries of geniuses.