Question

Find whether or not the following differential equation is exact. if exact, then solve the differential equation: (y^(2 ) e^(xy^2 )+4x^3 )dx + (2xye^(xy^2 )-3y^2 )dy=0

Answer 1

Answer:

We can know at the start if it is an exact equation or not!

Imagine we do these further partial derivatives:

∂M∂y = ∂2I∂y ∂x

∂N∂x = ∂2I∂y ∂x

they end up the same! And so this will be true:

∂M∂y = ∂N∂x

When it is true we have an an "exact equation" and we can proceed.

And to discover I(x, y) we do EITHER:

I(x, y) = ∫M(x, y) dx (with x as an independent variable), OR

I(x, y) = ∫N(x, y) dy (with y as an independent variable)

And then there is some extra work (we will show you) to arrive at the general solution

I(x, y) = C