A strange differential equation y''*(y-1)=y'

TL;DR

The video demonstrates how to solve a challenging differential equation, where the solution cannot be expressed in terms of x but can be expressed in terms of y.

Transcript

okay let's do some fo fun here welcome solve this differential equation here we have y double prime times apprentices with y -1 inside and that's equal to Y Prime and we're going to solve this when y is greater than 1 well notice that if whites exactly equal to 1 the left hand side is just going to be 0 and when we differentiate one we get 0 namely... Read More

Key Insights

• ➿ Solving the given differential equation involves isolating y double prime and integrating both sides with respect to x.
• 🆘 Introducing a substitution helps simplify the equation and make it easier to integrate.
• 🍉 The exponential integral function is used to solve the equation in terms of y.
• ❣️ The solution obtained is x as a function of y, rather than y as a function of x.
• ❓ The differential equation cannot be solved using elementary functions.
• 🎮 The video was inspired by a question from a follower on Twitter.
• ❓ The solution to the differential equation involves a constant, which is determined through integration.

Questions & Answers

Q: Can the given differential equation be solved in terms of y and x?

No, the solution cannot be expressed in terms of y but can be expressed as x as a function of y.

Q: What are the steps involved in solving the differential equation?

The steps include isolating y double prime, integrating both sides, introducing a substitution, solving for dy, integrating again, and using the exponential integral function.

Q: Why is isolating y double prime an important step?

It allows for the equation to be transformed into an equation in terms of y, making it easier to integrate both sides.

Q: What is the role of the exponential integral function in solving the differential equation?

The exponential integral function is used to integrate the equation in terms of y, ultimately providing the solution as x as a function of y.

Summary & Key Takeaways

• The video provides step-by-step instructions on solving a differential equation involving y double prime, y prime, and y when y is greater than 1.

• By isolating y double prime and integrating both sides, the equation can be transformed into an equation in terms of y.

• The equation can then be solved using a special function called the exponential integral function to find x as a function of y.