Help on first order differential equation!

Solve

Cosx(dy/dx) + ysinx = cosxsin2x , y(0) = 1

Can I get a step by step guide as to how to solve this? I've kind of forgotten how to,

What I've done so far:
Get into form: dy/dx + P(x)y = Q(x)

So I have:
dy/dx + ytanx = sin2x

Integrating factor is: e^ Int(P(x))
= 1/cosx
(skipped the calculations here)

Where do I go from here? A step by step guide would be much appreciated!!

Thank you

Integrating factor = I

$Iy = \int IQ(x) dx$

Why do we ignore the P(x)y ?

You can use the chain rule to find d/dx(y/cosx) if you want to show yourself that it works.

*Product rule.

Once you have the integrating factor then really what you want do is to multiply both sides of the equation by it, in your case to get:
(1/cosx)*dy/dx + y*(1/cosx)*tanx = sin2x/cosx

The reason this is useful is because the LHS is now actually (y/cosx)' (ie. I(x)*y)

You can then integrate both sides with respect to x (the LHS will just be y/cosx, the RHS looks a bit nasty but I'll leave that to you!)

Why does the LHS become y/ cosx ?

Sorry- that was (y/cosx)' which was my notation for the differential of y/cosx with respect to x.
This is a case where it isn't immediately obvious it can be written in this form but you can differentiate this to check (and with integrating factor, getting the left hand side as the differential of something is always what you are trying to do and the method will always deliver this).

The reason for the method is below:

We want to change the equation of the form dy/dx + P(x)y=Q(x) into something of the form
d/dx (something*y) = ....
so that we can then integrate both sides, and to do this we will multiply both sides by an "integrating factor", call it R(x).

This would give us R(x)dy/dx + R(x)P(x)y = R(x)Q(x) [Equation 1]

Using the product rule, d/dx (R(x)y) = R(x)dy/dx + dR/dx * y [Equation 2]

Comparing the LHS of Equation 1 with the RHS of Equation 2, they will be the same if R(x)*P(x) = dR/dx
Using the method of separating variables, you can solve this to get
R(x)=e^Int(P(x))

Think that was as good as explanation gets except the bolded is wrong. Assuming a typo

Thanks, but am I being thick here- what is wrong about the bit in bold? The integrating factor should be e to the power of the integral of P(x)? No?