Solve the Separable Differential Equation y'(t) = e^(y/2)sin(t) | Summary and Q&A

TL;DR

The video walks through the process of solving a differential equation using integration and substitution.

Key Insights

• 💁 Differential equations can be rewritten in a more manageable form to facilitate solving.
• 🙃 Separating variables and integrating both sides is a crucial step in solving differential equations.
• 🥺 Making substitutions can simplify the integration and lead to faster solutions.

Transcript

hello in this video we're going to solve this differential equation let's just go ahead and jump into it right away and start the process so solution the first thing I want to do is write this differential equation in a way that is just a little bit easier to solve so we can write y Prime of t like this d y DT and then over here on the right hand s... Read More

Questions & Answers

Q: How does the video suggest rewriting the given differential equation?

The video suggests representing y' as dy/dt and rewriting the equation with all y terms on one side and all t terms on the other.

Q: What is the trick used to integrate the sine of t?

The trick is to make a substitution by letting u equal to negative y/2 and then multiplying both sides by -2 to eliminate the coefficient in the integrand.

Q: What is the solution to the differential equation?

The solution is given as -2e^(-y/2) = -cos(t) + C, where C represents the constant of integration.

Q: What is the shortcut for integrating e^(-y/2)?

The shortcut is to divide the integrand by the coefficient (-1/2), resulting in -2e^(-y/2), as stated in the solution.

Summary & Key Takeaways

• The video demonstrates how to rewrite a differential equation to make it easier to solve.

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

• A substitution is made to simplify the integration step, and the solution is found by applying integration formulas.