00:01
So the question asks us to find the general solution of a differential equation, which is y prime, which equals to 2x over y times by e to the power of y minus x.
00:15
So we can start off with rewriting y prime into d y over d x.
00:22
And here in this case, we're also going to use the exponential law, which is e to the power of a minus b, is identifiable.
00:31
To e to the part of a over e to the part of b.
00:36
So we're going to write this as e to the part of y over e to the power of x.
00:41
And now because these are separable equations, we're going to put all the x's on one side and all the y's on the other side.
00:48
So our equation becomes y over ey, dy, and 2x over ex dx.
00:55
Now we will need to integrate both sides of the equation.
00:59
And in this case, the left -hand side needs to be solved using integration by parts.
01:05
So the integration by parts formula is integral of fg prime equals to fg minus the integral of f prime g.
01:17
So we're going to let our f equal to y and our g prime to equal to 1 over ey.
01:25
Now then our f prime just simply becomes 1 and then we need to find, we need to do an integral to find our g.
01:33
So it simply is 1 over ey, but there is a minus at the front because ey is in the denominator, so it has an exponent of negative 1.
01:45
And then now all we need to do is go fg, so f times g, so y minus ey minus f prime g...