Find equations for the flow lines, <1/y, 2x>

I'm wondering if someone could show me how to check this, here's my work:

Use a first order differential equation to solve:

dy/dx = 2x/(1/y)

(1/y)(dy/dx) = 2x

integrate with respect to x:

∫ (1/y)(dy/dx) dx = ∫ 2x dx

ln y = x^2 + c

How would I go about checking this answer for accuracy?

Ok, that's correct.

You could also solve for y as a function of x:
y = e^(x^2 + c)
y = e^(x^2) * e^c

Now since c is an arbitrary constant, then so is e^c.
We can then replace e^c with C, another arbitrary constant.

y = C e^(x^2)

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Check:

dy/dx = C * 2x * e^(x^2) = 2Cx e^(x^2)

2x/(1/y) = 2xy = 2x C e^(x^2) = 2Cx e^(x^2)

dy/dx = 2x/(1/y)

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Now if you wish to leave as

ln y = x^2 + c

we can check in similar fashion.
We use implicit differentiation to find dy/dx

1/y dy/dx = 2x
dy/dx = 2x/(1/y)