00:01
For this problem, we are asked to find the particular solution to the differential equation d -u by d -v equals u -v -s -s -v -squared that satisfies the initial condition u of 0 equals 1.
00:12
So the first thing that we can do here is divide both sides by u.
00:16
So we'd have d -u over u and multiply both sides by d -v.
00:21
So we have d -u over u equals v times sine of v squared.
00:28
Now, what we can do further is make the substitution.
00:34
Okay, actually, i'm getting ahead of myself.
00:37
What we can do on the left -hand side there is integrate with respect to u, which is going to give us lawn of you.
00:46
And on the right -hand side, we can integrate, oh, the right -hand side is going to require a little bit more intense process.
00:54
So the integral of v -sign -v squared, dv, oops not du dv we can do using u substitution or i'll call it x substitution so we'll have x equals v squared so d x is going to equal 2v dv which then means that dv is going to equal dx over 2v so making that substitution we'll turn the integral on the right hand side there into 1 over 2 times the integral of well the vs will multiply out, so we are just left with 1 over 2 times the integral of sine of x, dx.
01:35
Now the integral of sine of x, dx, if we differentiate cos, it goes to negative signs...
