Exact Differential Equation (x^2y^3 - 1/(1 + 9x^2))dx/dy + x^3y^2 = 0 | Summary and Q&A

TL;DR

The video explains how to solve a differential equation by checking if it is exact and then integrating the separate parts with respect to their respective variables.

Key Insights

• 💁 The differential equation is converted into the form Mdx + Ndy = 0 to check if it is an exact differential equation.
• 🆘 The computation of partial derivatives helps determine if the equation is exact or not.
• 🫡 Integrating each part of the equation with respect to their respective variables helps retrieve the unknown function and solve the differential equation.
• 🎮 The formula for integrating 1/(a^2 + x^2) is used in the video to integrate a specific part of the equation.

Transcript

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Questions & Answers

Q: How do you determine if a differential equation is exact?

To determine if a differential equation is exact, you need to compute the partial derivatives del M del Y and del N del X and check if they are equal. If they are equal, the equation is exact.

Q: What is the purpose of integrating each part of the equation separately?

Integrating each part of the equation separately allows you to retrieve the unknown function F. By integrating the first part with respect to X and the second part with respect to Y, you can set them equal to each other and solve for F.

Q: How do you integrate the second part of the equation that involves 1/(1 + 9x^2)?

To integrate the second part of the equation, you can use the formula 1/(a^2 + x^2) = 1/a * arctan(x/a) + C. By applying this formula and using a u-substitution, you can integrate the second part with respect to X.

Q: Why is it necessary to set the final answer equal to C?

Setting the final answer equal to C is important because it represents the constant of integration. This allows flexibility in the solution and accounts for any possible additional unknown functions.

Summary & Key Takeaways

• The video starts by putting the given differential equation in the form Mdx + Ndy = 0 and checks if it is an exact differential equation.

• The necessary partial derivatives are computed to determine if the equation is exact.

• If the equation is exact, the video explains how to solve it by integrating each part separately and setting them equal to each other.