You have gotten the DE to the point where
and so the integrating factor is .
Multiplying through by the integrating factor gives
.
Plugging in the initial condition gives
.
Therefore .
(1+x)y' + y = cos(x).... y(0) = 1
So far, I have:
P(x) = 1/(1+x)
Q(x) = cos(x)/(1+x)
p(x) = e^int(P(x))dx = e^int(1/1+x)dx = e^(ln(1+x))
... I think I did the integral wrong. It's been half a year, my integration skills need to worked on. But this homework is due in an hour and if somebody could real quickly guide me thru it and then I can work thru this section on my own when I don't have pressure to turn homework in... I would really appreciate it!
Once I have p(x), I multiply both sides of DE by it.
Then I'm supposed to "recognize" that left side is a derivative, so taking integral of both sides just means taking integral of right side.
Once I do that, I'll have a C, so is that when I plug in the initial condition?
And then once I get C and substitue that number in, that's my particular solution?
Thanks!