Question 827885
The easiest way to do this is to use implicit differentiation. (I will do with without implicit differentiation later.)<br>
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!<br>
Using "regular" differentiation (the quotient rule) on y = x/u, we get:
{{{dy/dx = (u(1) - x(du/dx))/u^2}}}
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/dx = u/u^2 - (x(du/dx))/u^2}}}
which simplifies to:
{{{dy/dx = 1/u - (x(du/dx))/u^2}}}
Factoring the second fraction:
{{{dy/dx = 1/u - (x/u)*(1/u)(du/dx)}}}
Since y = x/u we can substitute in for the x/u:
{{{dy/dx = 1/u - y*(1/u)(du/dx)}}}
Multiplying both sides by u (to eliminate the fractions):
{{{u*(dy/dx) = 1 - y*(du/dx)}}}
Adding y*(du/dx) to both sides:
{{{y*(du/dx) + u*(dy/dx) = 1}}}
And we're done!