Solving dy/dx=x-y | Learn how to integrate y without respect to anything

[computerex]

Homework Statement

\int (x-y)

What is \int y ? I don't mean \int y dy.

The Attempt at a Solution

dy/dx = x-y
y + dy = x dx
\int y + \int dy = \int x dx
\int y + y = x^2/2 + c

I am stuck at this point because I don't know how to integrate y without respect to anything...If that even makes sense.

EDIT:

Nvm...I am stupid xD

\int x dx - \int y dx

 

[gabbagabbahey]

EDIT:

Nvm...I am stupid xD

\int x dx - \int y dx

That still doesn't help you though. How exactly do you plan on integrating \int y(x) dx when you don't know what y(x) is?

You can't solve this differential equation just by integrating both sides. Instead, try using the substitution u=x-y to rewrite the DE in terms of u(x) and u'(x).

 

[D H]

Instead, try using the substitution u=x-y to rewrite the DE in terms of u(x) and u'(x).

That will result in another nonhomogeneous ODE. A tiny bit simpler perhaps, but still nonhomogeneous.

computerex: What have you been taught regarding solving nonhomogeneous differential equations?

 

[gabbagabbahey]

That will result in another nonhomogeneous ODE. A tiny bit simpler perhaps, but still nonhomogeneous.

Something about separable ODE's appeals to me though

 

[D H]

Depends on the OP's background. The original problem can be rewritten as

\frac{dy}{dx} + y = x

The homogeneous and particular solutions can be read off just by inspection.