Finding the stream function given velocity components

[Quinn Pochekailo]

Homework Statement

The velocity components in a steady, incompressible, 2-D flow field are

u = 2y
v = 4x

Find the corresponding stream function.

Homework Equations

u = ∂Ψ/∂y
v = -∂Ψ/∂x

The Attempt at a Solution

I can verify that a stream function exists for this problem because conservation of mass is satisfied. (∂u/∂x + ∂v/∂y = 0)

After plugging in my values for u and v, I get that Ψ = y² + f(x) and Ψ = -2x² + f(y).

I then equate the 2 equations to give me f(x) + y² = -2x² + f(y).

I then set f(y) to 0, to give me that f(x) = -2x² - y².

I then put f(x) back into my original Ψ equation to give me that Ψ = -2x² and this is the function that I get for my stream function.

However, my book is telling me that Ψ = -2x² + y² should be my stream function.

Am I doing something wrong?
Thanks in advance.

 

[Chestermiller]

If you take the partial derivatives of your stream function, do you get the correct velocities?