How to Solve the Partial Differential Equation u_xx = 0

TL;DR

Learn how to solve a basic partial differential equation by integrating and adding unknown functions of the variables.

Transcript

hey what's up YouTube in this problem I'm going to show you how to solve a very simple partial differential equation so in this problem you here is a function of two variables so you can actually be written as U of X comma Y kind of like you would see in a calculus 3 class so it's a multi variable function so use of X this denotes the first partial... Read More

Key Insights

• 🥅 Partial differential equations involve functions of multiple variables, and the goal is to find functions that satisfy the equation.
• 🫡 Integrating with respect to one variable can eliminate derivatives in the equation.
• 👻 Adding an unknown function of the other variable accounts for the integration process and allows for more possible solutions.
• ✖️ The solution to the partial differential equation includes a constant function multiplied by one variable and an unknown function of the other variable.
• ❓ Understanding the concept of treating one variable as constant when integrating is crucial for solving partial differential equations.
• ❓ The solution involves finding unknown functions that satisfy the given equation.
• ☺️ The notation "U sub X" represents the first partial derivative of U with respect to X.

Questions & Answers

Q: What does the notation "U of X, Y" represent in the partial differential equation?

"U of X, Y" represents a multi-variable function, where U is a function of both x and y.

Q: How is the rate of change of U in the x direction denoted?

The rate of change of U in the x direction is denoted as "del U del X" or "U sub X."

Q: Why is an unknown function of y added when integrating with respect to x?

When integrating with respect to x, the derivative of the unknown function of y with respect to x is zero. To include this possibility, the unknown function is added.

Q: What is the solution to the partial differential equation?

The solution is given by U of XY = F(Y) * X + G(Y), where F(Y) is a constant function of Y and G(Y) is an unknown function of Y.

Summary & Key Takeaways

• The content explains how to solve a simple partial differential equation with two variables, x and y.

• By integrating the equation with respect to x and adding unknown functions of y, the solution can be found.

• The solution includes a constant function of y multiplied by x and an unknown function of y.