Please describe what is INTEGRATION FACTOR in a linear differential equation. Please explain with a worked example. Thanks.

example => dy/dx + y/x = x...................1

this equation is similar to dy/dx + y(f(x)) = g(x)            .....................2

these types of differential equations are solved by following steps

at first calculate I.F (integration factor)

I.F = e^f(x)dx

solution of this equation 2 will be now

y(IF) = g(x)IF dx                     ........................3

for equation 1 , IF = e^f(x)dx = e^(1/x)dx = e^lnx = x

solution will be

yx = x²dx

yx = x³/3 + c

if u still having problem then ask