In Exercises 1–22, solve the differential equation.

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Question

In Exercises 1–22, solve the differential equation.

y' = eʸ/xy

Accepted Answer

Hello. In this video, we are going to be finding the general solution of the given differential equation. Now, in order to find the general solution of the differential equation, we can go ahead and use the method of separation of variables. So the first thing we would want to do is you want to go ahead and rewrite Y in the form of DYDX. So we could take this equation and rewrite it as DYDX is equal to x multiplied by e to the power of Y, all divided by y. Now, what we want to go ahead and do is we want to keep all the Y terms to the left hand side of the equation and keep all the X terms to the right hand side of the equation. So, we will go ahead and multiply by the reciprocal of e to the power of Y divided by y, and we will multiply both sides of this equation by DX. So we will be left with Y divided by e to the power of Y, DY equal to XDX. Now, the next thing we want to do is you want to go ahead and remove the differential symbols D Y D X. In order to do this, we will need to integrate both sides of this equation. So Let's go ahead and start by integrating the left-hand side. Now the left hand side is going to require integration by parts. So let's go ahead and do that work on the side here. So we have Y multiplied by e to the power of negative YDY. Now, in this case here, let's go ahead and set U equal to Y, and we're going to set DV to equal toe to the power of Y. By taking the derivative of u, we will have duu is equal to Dy, and by integrating the dv statement we will have v is equal to negative e to the power of negy. Now recall that integration of pi parts can be written as u multiplied by V minus the integral of V multiplied by DU. So here, in order to integrate by using integration by parts, we multiply U and V together, and that is going to give us ye to the power of Y minus the integral of V multiplied by DU, which is e to the power of YDY. Now here we have a constant of -1. Let's go ahead and move that to the front of the integral. That will allow us to rewrite the integral as ye to the power of y plus the integral of e to the power of YDY. And now if we were to integrate the exponential integrand, that will leave us with ye to the power of Y minus EY plus a generic constant C. So, that is going to be the integral of the left hand side. In fact, what we can go ahead and do is we can even factor on E to the negative Y, and that is going to leave us with negative Y. -1 multiplied by e to the power of y plus c. And if we were to further multiply negative or factor out a -1 from the group, we get Y + 1 multiplied by e to the power of Y + C. So that's going to be the work for the left hand side. Now for the right hand side. The right hand side, we just have the integral of X, and the integral of X is going to integrate to be 1/2 X2 + C. Now, because we have two generic constants, we are going to label one as C1 and the other as C2. Now, if we were to plug this back into the equation, we get Y + 1 multiplied by EY + C1 equal to 1/2 X2 + C2. Now, we can go ahead and combine the generic constants C1 and C2 together on the right hand side, and name that as a new generic constant C3. So we get Y + 1 multiplied by E to the power of Y equal to 1/2 X2 + C3. And now, what we want to go ahead and do is we just want to remove the -1 coefficient from the left hand side, so we can solve for Y. By doing so, we divide everything by -1 and we are left with y + 1 multiplied by e to the power of y equal to c minus 1/2 x 2. Now, in this case here, it is a little bit difficult to simplify Y + 1 multiplied by E to the power of Y. So we can go ahead and leave our solution in this form, but this is going to be the overall solution to the problem. And the correct option for this problem is going to be option A. So I hope this video helps you in understanding on how to approach this problem, and we will go ahead and see you all in the next video.

In Exercises 1–22, solve the differential equation.y' = eʸ/xy

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In Exercises 1–22, solve the differential equation.

y' = eʸ/xy