Question

solve xdy/dx-y=-x

Answer 1

First order linear differential equation.
Put it into standard form first by dividing everything by x.

Answer 2

You're first step is to find the integrating factor u(x)

Answer 3

sorry p(x)*

Answer 4

idk if you're there still or not to proceed explaining this...

Answer 5

but your integrating factor is: \[e^{-\ln|x|} = \frac{ 1 }{ x }\]
I'm going to skip the work for the next few steps, you can use it as a reference to absolutely check your work, but you SHOULD see that you get product rule as a result.
Therefore: \[[\frac{ 1 }{ x }y]'=-1\]
Next, take the integral: \[\int\limits [\frac{ 1 }{ x }y]'= - \int\limits dx = \frac{ 1 }{ x }y= -x+C\]
\[y = -x^2+Cx\]

Answer 6

the equation should be \[\Large \frac{1}{x}y=-\int\frac{1}{x} (1) dx\]

Answer 7

@sirm3d could you show him how you got that?

Answer 8

the differential equation is \[y'-\frac{1}{x}y=-1\]
with integrating factor \(1/x\)

the resulting equation is
\[(1/x)y=\int(1/x)(-1)dx\]