00:01
Hey there, in this video we'll find a solution to sine y times the quantity x plus sine y d x plus 2x squared cosine y d y equals 0.
00:08
The first thing we want to do this equation is simplify a little bit by performing a substitution u in for sine of y.
00:15
This also implies that du is cosine of y d y d y, so our equation becomes u times x plus u d x plus 2x squared u, ours or 2x squared d u equals 0.
00:26
And this we can recognize as a bernoulli equation.
00:30
This is great because this is an kind of equation that we know how to solve.
00:34
Just as a reminder, a bernoulli equation is an equation of the form.
00:39
Y prime plus p of x y equals q of x y to the n.
00:43
And we can solve this by performing the substitution z equals y to the 1 minus n as long as n is not equal to 0 or 1.
00:52
In the case that n is 0, we're going to get a linear equation for this thing instead, and we can just use the usual methods for linear equations.
00:59
And if n is equal to 1, this is a separable equation, so we can solve it that way anyway, but we only use the substitution in the case that n is not equal to 0 or 1.
01:09
Otherwise, the substitution doesn't make a lot of sense.
01:12
So i'd like to rewrite the equation that we have here in this sort of bernoulli equation form.
01:17
Of course, u is going to be playing the role of y for us.
01:20
To start out, i'm going to divide everything by dx and also by 2x squared.
01:25
So my expression becomes, u over x plus x plus u over 2x squared plus d u x equals 0.
01:33
D u or d x is just u prime so i can rewrite this whole thing as u prime plus if i distribute the division over the addition terms u prime plus 1 over 2x u plus 1 over 2x squared u squared equals 0 and then lastly i'm just going to move that second term over to the right hand side and we have the equation u prime plus 1 over 2 x u minus 1 over 2x squared u squared equals minus 1 over 2x squared u squared and this is exactly of the form of a bernoulli equation up above where here my p is 1 over 2x my q is negative 1 over 2x squared and my n is 2 which is not 0 or 1 so i'm very happy about that now just as it says in order to solve this equation we're going to do the substitution z equals 1 minus i is y to the 1 minus n y here is actually u because we've done that substitution up above and n here as we see is that 2.
02:29
So we're substituting z equals u to the negative 1.
02:33
This also implies that z prime is equal to negative u to the minus 2 u prime by the chain rule.
02:42
So substituting n for z prime r for u prime.
02:46
I'm going to get u prime here is equal to negative u squared z prime.
02:54
So putting that into the original equation, negative u squared z prime, plus 1 over 2x u equals negative 1 over 2x squared u squared.
03:06
This equation is currently just a mess because it's involving u's and x's and z's, but we're going to remedy that right away.
03:12
First thing we're going to do is divide everything out by that common factor of u, and i have negative u z prime plus 1 over 2x equals negative 1 over 2x squared u.
03:23
Now, using the fact that u is 1 over z, i can replace you everywhere with 1 over z, and i end up with the equation negative 1 over z times z prime plus 1 over 2x equals negative 1 over 2x squared times 1 over z...