### Video Transcript

Find the implicit solution to the following differential equation. The sin of ๐ฆ multiplied by d๐ฆ by d๐ฅ minus the cos of ๐ฅ is equal to zero.

The question gives us a differential equation, and it wants us to find the implicit solution of this differential equation. Remember, when weโre asked to find an implicit solution, we donโt need to give our answer as a function of ๐ฅ. So letโs try and find the solution to our differential equation. The first thing we notice is weโre given a function of ๐ฆ and a function of ๐ฅ in our differential equation. Since thereโs no way of manipulating our equation to remove our function in ๐ฆ, we should try writing this as a separable differential equation. In fact, in this case, we can just skip the step where we write this as a separable differential equation. Weโll just add the cos of ๐ฅ to both sides of the equation.

Doing this, we get the sin of ๐ฆ multiplied by d๐ฆ by d๐ฅ is equal to the cos of ๐ฅ. And now we want to separate our variables. We have our function of ๐ฆ on the left-hand side of our equation and our function of ๐ฅ on the right-hand side of our equation. And to solve this, we need to use a trick which we use when weโre solving separable differential equations. We know d๐ฆ by d๐ฅ is not a fraction. However, we can treat it a little bit like a fraction when weโre solving separable differential equations. This gives us the equivalent statement the sin of ๐ฆ d๐ฆ is equal to the cos of ๐ฅ d๐ฅ. And we can then solve this by integrating both sides of our equation. We get the integral of the sin of ๐ฆ with respect to ๐ฆ is equal to the integral of the cos of ๐ฅ with respect to ๐ฅ.

Weโre now ready to evaluate both of these integrals. First, to integrate the sin of ๐ฆ with respect to ๐ฆ, we recall the integral of the sin of ๐ with respect to ๐ is equal to negative the cos of ๐ plus a constant of integration. So by calling our variable ๐ฆ, we get that this integral was equal to negative the cos of ๐ฆ plus a constant of integration weโll call ๐ถ one. Now, to evaluate the integral of the cos of ๐ฅ with respect to ๐ฅ, we recall the integral of the cos of ๐ with respect to ๐ is equal to the sin of ๐ plus a constant of integration. This time, since weโre integrating the cos of ๐ฅ with respect to ๐ฅ, weโll label our variable ๐ฅ. This gives us the sin of ๐ฅ plus a constant of integration weโve called ๐ถ two.

At this point, we try and write this in the form ๐ฆ is equal to some function of ๐ฅ. But the question is telling us to find an implicit solution, so we donโt need to do this step. Instead, we can rearrange our solution into any form which we want. Weโll start by combining our constants ๐ถ one and ๐ถ two into a new constant on the left-hand side of our equation weโll call ๐ถ. Next, weโll add the cos of ๐ฆ to both sides of this equation. This gives us ๐ถ is equal to the sin of ๐ฅ plus the cos of ๐ฆ. And weโll rearrange this equation one more time. Weโll rearrange this to get the cos of ๐ฆ plus the sin of ๐ฅ is equal to ๐ถ.

Therefore, given the differential equation the sin of ๐ฆ times d๐ฆ by d๐ฅ minus the cos of ๐ฅ is equal to zero, we were able to find the implicit solution the cos of ๐ฆ plus the sin of ๐ฅ is equal to ๐ถ.