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## Linear Differential Equation with a substitution.

Wow I am so very bad at differential equations.. :(

The problem

Here is the exact problem I'm given:

attempt at a solution

I'm guessing that I need to differentiate y(x) that I am given and substitute that into the left hand side and then put the y function [sinx + 1/u(x)] as y in the right hand side (y^2).
Then re-arrange for u(x) and differentiate to get du/dx and hopefully it will be the third given equation.

After that I think I just have to solve that ODE and get it back in terms of y.

Assuming that is what I have to do, sadly I cannot even get that far..

differentiating y(x) = sinx + 1/u(x)
must this be done implicitly? I get confused as u(x) is a function and not a variable.

I can see that the third equation must be solved linearly such that dy/dx + utan(x) = -1/2 sec(x)

And I (think) that I solved the integrating factor to be 1/cos(x).

But I am so bad at differential equations and the sources on the internet seem to be so confusing that I'm quite stuck from here but this is what I have:

(1/cos(x))*du/dx + u*(1/cos(x))*tan(x) = -(1/2)*(1/cos(x))*sec(x)=d/dx(u/cos(x)) <- multiplying by integrating factor and copying form of examples from the net

u/cos(x) = integral of -(1/2)*(1/cos(x))*sec(x)

and I know I'm way off here so I think I might aswell just stop...

Also it would be great if someone could point me in the direction of a site that shows how to type all those math symbols such as the integral sign.

Thanks in advance. A lot.

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