Students find hardships in differential equations because the subject requires strong mathematical skills, a deep understanding of calculus, and the ability to visualize and manipulate abstract concepts. Additionally, the complexity and variety of problems encountered in differential equations can present significant challenges.

## So let us investigate the query more attentively

Differential equations are notorious for being a challenging subject for students because of their fundamental complexity and practical applications. As stated previously, this requires strong mathematical skills, a deep understanding of calculus, and the ability to visualize and manipulate abstract concepts. However, the challenges of differential equations go beyond these technical requirements.

As Stephen Hawking once famously said, “It is not enough to be able to solve problems. One must also know which problems to solve.” This applies directly to differential equations, as students must not only understand the mathematics behind the subject but also how to apply it to real-world problems. Moreover, the variety of problems encountered in differential equations can present significant challenges, as there are countless ways to approach a single problem.

Here are some interesting facts about differential equations:

- Differential equations have been used to model everything from chemical reactions to financial markets.
- Differential equations are closely related to analytic geometry, and both subjects rely heavily on calculus.
- The study of differential equations has led to numerous advances in engineering, physics, and computer science, among other fields.
- Many famous mathematicians have contributed to the field of differential equations, including Isaac Newton, Leonhard Euler, and Bernhard Riemann.

To help illustrate the complexity and importance of differential equations, here is an example of a table that shows some common types of differential equations and their applications:

Type of Differential Equation | Example Application |
---|---|

Ordinary Differential Equation | Modeling population growth |

Partial Differential Equation | Modeling heat transfer in materials |

Stochastic Differential Equation | Predicting stock prices |

Nonlinear Differential Equation | Describing chaotic systems |

In conclusion, differential equations are a challenging subject that requires a deep understanding of mathematics and real-world applications. By understanding the broader significance of this subject, students can better navigate the complexity and develop the necessary skills to excel.

## On the Internet, there are additional viewpoints

I’d say it’s because people don’t really study and memorize the methods and techniques to solving Ordinary Differential Equations and therefore, when it comes to a quiz or an exam they struggle to figure out how to approach solving the differential equation. Now, if we’re talking Partial Differential Equations then people struggle because it’s just flat out hard as hell just by itself, P.D.E.’s can be extremely complex and tedious due to the amount of independent variables involved which in turn requires more in-depth understanding of differentiation techniques specifically partial differentiation, and a lot of times the equations don’t even have general solutions, and it’s because of this that P.D.E.’s are so difficult.

## Answer in the video

In this video, the speaker emphasizes the importance of learning the five methods for solving differential equations, using the example of a simple harmonic oscillator. The five methods include making a guess and checking, using energy conservation, using series expansion, using Laplace transforms, and visualizing the behavior of the solution as a flow on face space using Hamilton’s equations. The speaker provides examples and thorough explanations for each of the methods, and notes that these tools become invaluable as physicists encounter more complex differential equations. The video creator also thanks their supporters on Patreon and highlights the use of the manim programming library for math animations.