SOLUTION: Suppose y=x/u, where u is a function of x. Show that y(du/dx)+ u(dy/dx)=1 Thanks!!

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Question 827885: Suppose y=x/u, where u is a function of x. Show that y(du/dx)+ u(dy/dx)=1
Thanks!!
Answer by jsmallt9(3758) About Me  (Show Source):
You can put this solution on YOUR website!
The easiest way to do this is to use implicit differentiation. (I will do with without implicit differentiation later.)

If we multiply each side of the equation by u we get:
y*u = x
Implicitly differentiate. On the left side we will use the product rule:
y*(du/dx) + u*(dy/dx) = 1
And we're done!

Using "regular" differentiation (the quotient rule) on y = x/u, we get:
dy%2Fdx+=+%28u%281%29+-+x%28du%2Fdx%29%29%2Fu%5E2
Now we use some algebra to transform this equation into the desired one. Since the desired equation has two terms on one side, I will split the fraction (i.e. "un-subtract"):
dy%2Fdx+=+u%2Fu%5E2+-+%28x%28du%2Fdx%29%29%2Fu%5E2
which simplifies to:
dy%2Fdx+=+1%2Fu+-+%28x%28du%2Fdx%29%29%2Fu%5E2
Factoring the second fraction:
dy%2Fdx+=+1%2Fu+-+%28x%2Fu%29%2A%281%2Fu%29%28du%2Fdx%29
Since y = x/u we can substitute in for the x/u:
dy%2Fdx+=+1%2Fu+-+y%2A%281%2Fu%29%28du%2Fdx%29
Multiplying both sides by u (to eliminate the fractions):
u%2A%28dy%2Fdx%29+=+1+-+y%2A%28du%2Fdx%29
Adding y*(du/dx) to both sides:
y%2A%28du%2Fdx%29+%2B+u%2A%28dy%2Fdx%29+=+1
And we're done!