In Exercises 1–22, solve the differential equation.

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Question

In Exercises 1–22, solve the differential equation.

sec x dy + x cos² y dx = 0

Accepted Answer

Hello there, today we are going to solve the following practice problem together. So first off, let us read the problem and highlight all the key pieces of information that we need to use in order to solve this problem. Find the solution of the differential equation cosecant of xDy plus x multiplied by sec to the power of 2 of y dx is equal to 0. OK, so it appears for this particular prompt, we're ultimately asked to solve this particular differential equation. So we're asked to find the solution of this differential equation. So now that we know that we're ultimately trying to find or figure out what the solution of this differential equation that is given to us, our first step that we need to take. Is that based on our prior knowledge we will need to recall that in order to solve this specific differential equation we will need to recall how to use the separation of variables method by rearranging the terms to group y and x on opposite sides of our equation expression. And then we will need to integrate the resulting trigometric expressions by recalling and applying a power reduction identity for the Y integral, and an integration by parts method is what we'll need to recall and use to solve for the x integral. Finally, we will need to combine the results that we find, and we will need to simplify in order to express the general solution in its explicit or in its implicit, I should say specifically, because we ultimately want to express our general solution in its implicit form. So with that in mind, that's an overview of what we're going to do. Our first step is we need to begin the separation of variables. So we will need to rearrange our equation expression once again in order to group the y terms on the left side of our expression and the x terms on the right side of our expression. So we'll go ahead and write that Kosi can't of X so cosi can't. Of X D Y. Will be equal to drumroll, negative, so negative X multiplied by C2, so multiplied by C 2 Y. So C can't square of Y. D X OK, so, moving right along here, we now need to divide by c2 Y and multiply by sin of x, since as we should recall, let's make a note, we need to recall that sin of x is equal to 1 divided by cosecant. Of X. So that's an important little identity property we need to keep in mind. Fantastic. So, we can go ahead and write that 1 divided by C 2 YDY will be equal to X multiplied by sin of XDX. And we can simplify this expression to write cosine squared of YDY is equal to x multiplied by sin of xDX. Our second step now is we need to integrate both sides. So we need to set up the integrals for both sides. So the integral of cosine squared of y, D Y is equal to the integral of parentheses of x multiplied by sin of xDX. So now, our third step is we need to evaluate the y integral first. So we will need to, once again, we will need to use specifically recall and use the power reduction identity, which states that cosine squared of y will be equal to 1 plus cosine of 2y divided by 2. Fantastic. So we will be able to go ahead and write that the integral of 1 plus cosine of 2y divided by drum roll 2 multiplied by Dy will be equal to the integral of parentheses of 1/2 plus cosine of 2y divided by 2. DUI. And when we write our final answer in its most simplified form, I'm going to skip a few steps, but it's basic integration, we will find that it's simply equal to, when we write our answer once again in its most simplified form, 1/2 multiplied by y. Was 1/4 multiplied by drum roll. Sign up, so it's gonna be multiplied by sin of 2 Y, and then we need to add C subscript 1, which for simplicity's sake, I'll just call C1. Fantastic. So now our fourth step. As we now need to evaluate our x integral. So in order to solve for our x integral, like we mentioned earlier, we will need to use integration by parts for the integral of parentheses of negative x multiplied by sin of x dx. So, we will state that U will be equal to X, which will mean that DU by DX will be equal to 1, which will mean that DU will be equal to DX. And we will be able to write that DV will be equal to negative sin of xDX, which will mean that the interval of dV is equal to the interval of parentheses of negative sin of xDX, which will mean that V will be equal to cosine of x. Fantastic. So let me recall and apply the integration by parts formula, which states that the integral of UDV is equal to u multiplied by V minus the integral of V. D, we can go ahead and write that the integral of negative x multiplied by sin of x dx is equal to x multiplied by cosine of x minus the integral of cosine of x dx. Which can be written as the integral of x multiplied by sin of xDX, which will be equal to x multiplied by cosine of x minus sin of x plus C2, C subscript 2. And that's it So now for our 5th step is we need to combine our results, and when we equate both sides, it will give us the implicit general solution. So we need to write 1/22 multiplied by y plus 1/4 multiplied by sin of 2y plus c1 is equal to x multiplied by cosine of x minus sin of x plus c2. Fantastic. So, in order to clear the fractions, we will need to multiply the entire equation by 4, and we will need to combine and write the new integral constant as simply C C denotes a constant variable, that's what C represents. So that will mean that our final answer without a doubt will be 2 multiplied by Y 2 Y. Plus sin of 2 y is equal to 4 x multiplied by cosine of x minus 4 multiplied by sin of x plus c, and that's it. That is our final answer. Hooray, we did it. So, going back to look at the prom itself and to recap what we've learned, we could therefore safely say that the solution for our differential equation once again is 2y plus sin of 2y is equal to 4 x multiplied by cosine of x minus 4 multiplied by sin of x plus c, and that's it. Thank you so much for watching, hopefully that helped, and I can't wait to see you in the next video. bye.

In Exercises 1–22, solve the differential equation.sec x dy + x cos² y dx = 0

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In Exercises 1–22, solve the differential equation.

sec x dy + x cos² y dx = 0