Separable differential equation, ex2

TL;DR

Learn how to solve a differential equation by separating variables and integrating.

Transcript

okay we going to solve this differential equation we have x * y^ 2 * y Prime this is equal to x + 1 and by the way this is sequ dy DX right so our goal is first put all the X along with DX together on one side and then the Y and Dy together on the other side and since X is right here already and I don't want this x right here let's divide everythin... Read More

Key Insights

• ❓ Differential equations can be solved by separating variables and integrating.
• 🙃 Dividing both sides of the equation by a variable can help rearrange the equation.
• 🙃 Integrating both sides allows for finding the antiderivatives and isolating the variables.
• ❓ Constants (such as C1) can be introduced during the integration process and accounted for in the final solution.
• 🆘 Manipulating the equation can help simplify the solution.
• 🧊 The cube root function is used to undo the cubing operation on the variable y.
• ➕ The plus or minus sign is not needed when taking the cube root of both sides.

Questions & Answers

Q: What is the goal in solving this differential equation?

The goal is to rearrange the equation so that all the terms containing x and dx are on one side, while the terms with y and dy are on the other side.

Q: How does multiplying both sides by dx help in rearranging the equation?

Multiplying both sides by dx cancels out the dx term on the left side, allowing it to be moved to the right side of the equation.

Q: What are the next steps after rearranging the equation?

The next step is to integrate both sides of the equation, which involves finding the antiderivatives of the respective terms.

Q: How is the final solution obtained?

By isolating the variable y, the cube root of the expression 3x + 3ln|x| is taken. The solution is then written as y = (3x + 3ln|x|)^(1/3) + K, where K represents a constant.

Summary & Key Takeaways

• The content explains the process of solving a differential equation by separating variables and integrating.

• By dividing both sides by x and moving the dx term to the other side, the equation is rearranged.

• Integrating both sides allows for isolating the variable and finding the solution.