Take one side at a time.
Let f(x, y) = e^(x^2y).
[(df)/(dx)]_"left" = (e^(x^2y))[x^2(dy)/(dx)+y*2x]
(chain rule, product rule)
[(df)/(dx)]_"right" = 2 + 2*(dy)/(dx)
(chain rule with (dy)/(dx))
Now, set them equal...
(e^(x^2y))[x^2(dy)/(dx)+y*2x] = 2 + 2(dy)/(dx)
...Multiply through.
x^2e^(x^2y)(dy)/(dx) + 2xye^(x^2y) = 2 + 2(dy)/(dx)
You can now isolate (dy)/(dx). Move everything related to each other to each side.
2xye^(x^2y) - 2 = 2(dy)/(dx) - x^2e^(x^2y)(dy)/(dx)
Factor out common terms:
2xye^(x^2y) - 2 = (dy)/(dx)[2 - x^2e^(x^2y)]
Divide bracketed stuff over:
[2xye^(x^2y) - 2]/[2 - x^2e^(x^2y)] = (dy)/(dx)
...And simplify so it looks nicer.
-[2 - 2xye^(x^2y)]/[2 - x^2e^(x^2y)] = (dy)/(dx)
Wolfram Alpha lists the answer as:
[2 - 2xye^(x^2y)]/[x^2e^(x^2y) - 2] = (dy)/(dx)
