Question

x(dy/dx)+y=e^x

Answer 1

\[x(\frac{ dy }{ dx })+y=e ^{x}\]

Answer 2

what is it you need?

Answer 3

solving for anything?

Answer 4

it is a linear first order equation
coefficient of dy/dx is 1
coefficient of y is 1/x=p
R.H.S=q
soln is y*integrating factor=integral of (q*integrating factor) w.r.t. x
where integrating factor is exp(integral of p w.r.t x)

Answer 5

No integrating factor needed; the equation is ready to be solved as is:
\[xy'+y=e^x\\
\frac{d}{dx}\left[xy\right]=e^x\\
xy=\int e^x~dx\\
~~~~~\vdots\]