# Differential equation help

Homework Statement

Find the particular solution of the differential equation $$xdy=(x+y-4)dx$$ that satisfies the initial condition $$y(1)=7$$.

The attempt at a solution

Ok heres my first two steps...

$$dy = \frac{(x+y-4)}{x} dx$$
$$\frac{dy}{(x+y-4)} = \frac{dx}{x}$$

Now here's where I get messed up. How can I get the x out of the side with the dy? Could someone please explain how to finish this or If I'm headed in the right direction?

HallsofIvy
Homework Helper
Is "separation of variables" the only way you know to solve differential equations? This is NOT a "separable" equation.

Yea thats the only way I know how. But I'm looking at a different way now where they give you the standard equation y' +P(x)y = Q(x), but I don't quite understand it.

Yea thats the only way I know how. But I'm looking at a different way now where they give you the standard equation y' +P(x)y = Q(x), but I don't quite understand it.

THat's a linear first-order differential equation. Usually, textbooks show the method for solving them. Try looking in your textbook.

rock.freak667
Homework Helper
Use a substitution of y=Vx

Last edited:
tiny-tim
Homework Helper
y' +P(x)y = Q(x)

Homework Statement

Find the particular solution of the differential equation $$xdy=(x+y-4)dx$$ that satisfies the initial condition $$y(1)=7$$.
Yea thats the only way I know how. But I'm looking at a different way now where they give you the standard equation y' +P(x)y = Q(x), but I don't quite understand it.

Hi goaliejoe35!

Hint: first step: get the RHS x-only:

xdy - ydx = (x - 4)dx.

Does the LHS now remind you of anything?

If so, fiddle around with it until you get something you can integrate.

If not, go back to your book and look at y' +P(x)y = Q(x) again