Question

Can anyone help me solve for y? xy+(dy/dx) =100x

Answer 1

to clarify i can not find a way to separate the variables without being confident im doing the right thing

Answer 2

If
\[ xy + dy/dx = 100x \]
then
\[ \frac{dy}{dx} = x(100 -y) \]

Hence
\[ \frac{1}{100-y} \frac{dy}{dx} = x \]

Now integrate both sides with respect to x and we have
\[ \int \frac{1}{100-y} \frac{dy}{dx} dx = \int x \ dx \]

Notice on the left-hand side, that that integral can change variables to y, hence the LHS integral can be rewritten as
\[ \int \frac{1}{100-y} dy = \int x \ dx \]

Now integrate both sides and solve for y.

This technique btw, is called Separation of Variables, and is one of the most useful in solving elementary differential equations of the sort you were asked in this question.

Answer 3

\[\frac{dy}{dx}=100x-xy\]\[\frac{dy}{dx}=x(100-y)\]\[\frac{1}{100-y}dy=xdx\]\[\int\limits_{}^{} \frac{1}{100-y}dy=\int\limits_{}^{} xdx\]

Answer 4

ahhhh ty guys i dident even think about taking the x out in the second step. that makes a lot more sense.