Worked example: separable differential equation (with taking log of both sides) | Khan Academy

TL;DR

Learn how to solve a separable differential equation by integrating both sides and simplifying the expressions, resulting in an explicit function for y in terms of x.

Transcript

• [Instructor] Let's say we need to find a solution to the differential equation that the derivative of y with respect to x is equal to x squared over e to the y. Pause this video and see if you can have a go at it, and I will give you a clue. It is a separable differential equation. All right, now let's do this together. So, whenever you do any di... Read More

Key Insights

• ❣️ Solving a differential equation involves determining if it is separable, bringing all y and x terms to respective sides.
• 🔋 Multiplying both sides by suitable terms, such as e to the power of y and dx, assists in isolating variables.
• 🙃 Integrating both sides simplifies the equation by finding antiderivatives.
• 🔋 The integral of e to the power of y with respect to y is e to the power of y.
• ☺️ The indefinite integral of x squared with respect to x is x to the power of 3 over 3.
• 🙃 Adding a constant, represented by c, is necessary when integrating both sides.
• 🥘 Taking the natural logarithm allows for obtaining an explicit function for y in terms of x.

Questions & Answers

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

A differential equation is separable if you can isolate all the terms involving y on one side of the equation and all the terms involving x on the other side, with the differentials dy and dx on their respective sides.

Q: Why is it necessary to multiply both sides by e to the power of y and dx?

Multiplying both sides by e to the power of y eliminates the y term on the right side, allowing for separation of variables. Multiplying by dx gets rid of the dx on the left side and brings it together with the x squared term on the right side.

Q: What is the integral of e to the power of y with respect to y?

The integral of e to the power of y with respect to y is simply e to the power of y. This is possible because the derivative of e to the power of y with respect to y is also e to the power of y.

Q: How is the explicit function for y in terms of x obtained?

By taking the natural logarithm of both sides of the equation, the exponential e is canceled out from e to the power of y, resulting in y. Solving for y gives y = natural log (x to the power of 3 over 3) + c, where c represents an arbitrary constant.

Summary & Key Takeaways

• The video explains how to solve a separable differential equation, where the derivative of y with respect to x is equal to x squared over e to the power of y.

• To make the equation separable, both sides are multiplied by e to the power of y and dx, leading to e to the power of y dy = x squared dx.

• Integrating both sides allows for simplification, with the integral of e to the power of y with respect to y being e to the power of y.

• The indefinite integral of x squared with respect to x is x to the power of 3 over 3.

• Taking the natural logarithm of both sides results in y = natural log (x to the power of 3 over 3) + c, with c representing the constant.