Differential Equations: Rising complexity
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.
HOW MUCH MORE CHALLENGING CAN A 1ST ORDER LINEAR DIFFERENTIAL EQUATION GET?
A recap of the steps to solve a first order, linear differential equation in the form: 𝑑𝑦/𝑑𝑥+𝑝(𝑥)𝑦=𝑞(𝑥)
- Rearrange the differential equation to the form shown above
- Substitute 𝑦 as a product of 2 functions called 𝑢 and 𝑣, altering 𝑑𝑦/𝑑𝑥 as well
- Factor out the parts involving 𝑣 and equate this new expression to 0
- Use separation of variables and integration to find 𝑢
- Substitute this value into the equation of step 2
- Use that to find the 𝑣, and then use 𝑢 and 𝑣 to find the final solution 𝑦
The following example may seem very similar and even easier than the example in the previous post. Yet, be prepared to face more advanced concepts like integration by parts. [If the slides appear to be unclear, refer to the presentation using the link provided at the end of this post]
This is particularly hard example, hence it may take several attempts to simply understand. It may also be helpful to constantly switch between the previous post and the current one, referring to the similarities and differences. This concludes our journey with first order, linear differential equations.
Due to their rigorous solving methods, the vast majority of other differential equations will not be covered on this blog, especially partial differential equations. Unfortunately, these very partial differential equations are the only way to approach most theoretical physics concepts like heat, diffusion, electrostatics, electrodynamics, fluid dynamics, elasticity, gravitation and quantum mechanics.
Learn more about other types of differential equations, and the unique characteristics of every differential equation, through this oversimplified presentation.
hiii can u please post about ordinary differential equations and what solutions come if u solve them
ReplyDeleteHello!
DeleteOrdinary differential equations (ODE) are a particular set of differential equations which deal with derivatives with respect to only 1 variable {e.g. dy/dx, d^2y/dx^2, ...} [Exactly opposite are partial differential equations with equations that relate derivatives of multiple independent variables] {dy/dx and dy/dt, d^2y/du^2 and d^3u/ds^3,...}
The solution of any differential equation is a function or a set of functions. Even when dealing with just ODEs, the solutions can vary greatly. The most basic ODE (first order linear ODE) is discussed on the previous post, but I would recommend you to check out this handcrafted presentation that extensively covers the same topic.
https://drive.google.com/file/d/1pMhinJwyuGWOII58pzAsLC2OL-mzGyA3/view?usp=sharing