Hello all, this is my first time posting, and I need some help, big time!

Alright so my teacher gave us a homework assignment without much in the way of hints for how to approach the problems, so I would like to post the 3 problems here. The first two I have an answer for but would like verification that they are correct, and the last one I have no idea of how to approach.

1. Use Clairaut's Equation to show that the equation:

y = xy' + (y')^{2 }Has a singular solution that is the envelope of the one parameter family of solutions

Note: I am pretty confident about this one, so if you don't feel like spending the time with this one, I'd totally understandMy answer was:

General solution- y = c^{2}+cx

Particular Solution - y = -x^{2}/4

Because it involves a graph, I'll attach an image of the work.

2. Solve the equation dy/dx = y^{2}- 2/x^{2}

if it is known that y_{1}= 1/x is a particular solution of the equation. Hint: make y = z + 1/x

Okay so I used that substitution to transform the equation into a Bernoulli equation, and after a bit of arithmetic I got y = -3x^{2}/(x^{3}+c) + 1/x

The problem is that I do not understand where the y_{1}= 1/x comes into play. He wrote it just like that, with the 1 as a subscript (as opposed to y(1) = 1/x), so I just have no idea what to do with it.

I will again attach a picture in case you would like to follow my logic

3. Find the condition for which the equation:

(x-y)dx + (y+x)dy = 0

has an integrating factor of the form (lambda = u):

u = u(x^{2}+ y^{2})

also, find the integrating factor and solve the equation.

Okay... so I understand the concept of integrating factors and exact equations, but this just seems bizarre to me, what's with the two lambdas? Can anyone offer any guidance on this one?

I'll attach a picture of the test so that you could see exactly how it was written.

Any help that can be provided would be greatly appreciated, I'm getting a little desperate!

Thank you in advance!

Note: I had to zip all the pictures together because they were too large of a file.

Pictures.zip