How to Solve a Separable Differential Equation dy/dx = sec^2(y)/(1 + x^2)

TL;DR

This video demonstrates how to solve a separable differential equation using integration techniques.

Transcript

okay so we have a differential equation dy/dx equals the secant squared of Y over one plus x squared let's try to work this out solution so it appears that maybe we can separate the variables in other words we can get all of the X's on one side and all of the Y's on one side so to do that we'll divide both sides by secant squared and multiply both ... Read More

Key Insights

• 🙃 Separable differential equations can be solved by isolating the variables on different sides of the equation and integrating each side.
• ❎ The trig identity cosine^2(y) = (1 + cos(2y)) / 2 is useful for integrating functions involving cosine squared.
• 🗂️ When integrating trigonometric functions, dividing by the corresponding constant and applying the appropriate formula simplifies the calculations.

Questions & Answers

Q: What is the first step in solving the given separable differential equation?

The first step is to divide both sides of the equation by sec^2(y) to isolate the variables on each side.

Q: How is the left-hand side of the equation integrated?

The left-hand side is integrated using the trig identity cosine^2(y) = (1 + cos(2y)) / 2, which simplifies the integral to (1/2)dy + (1/2)∫cos(2y)dy.

Q: What is the trick to integrating functions involving trigonometric functions?

To integrate trigonometric functions, one should identify the corresponding derivative function (e.g., sine for cosine) and divide by the appropriate constant.

Q: What is the final step in solving the separable differential equation?

The final step is to integrate the function (1 + x^2)^(-1) with respect to x, which results in arctan(x) + C, where C is the constant of integration.

Summary & Key Takeaways

• The given differential equation is dy/dx = sec^2(y) / (1 + x^2).

• The content explains the process of separating the variables and integrating both sides of the equation.

• By applying a trig identity and integrating, the solution to the separable differential equation is obtained.