when im doing questions that use integrating factors i always face the same problem.
dy/dx + P(x)y = Q(x)
and the integrating factor, I, would be e^(integral of P)
however after multiplying the whole equation by I, you would normally get something like
Iy' + IP(x)y = IQ(x)
and the next step would be combining the left hand side into (###)' so that you could equate ### to integral of the right hand side. what i want to ask is how do you get from Iy' + IP(x)y to first derivative of ###?
and example is
dx/dt + 4x/(100-t) = 3/2
im not sure if its correct but i got
P = 4/100-t
=> I = (100-t)^-4 (correct this if im wrong)
then the equation becomes
x'(100-t)^-4 + 4x(100-t)^-5 = 3/2(100-t)^-4
what do i do from here? i always get stuck at this step for all problems using integrating factors.