$(40$ pts$)$ Consider steady, incompressible, laminar flow that is two$-$dimensional in the $x-y$ plane. Ignore gravity effects and units. The $x-$component of velocity is $u=-5x+3.$ Do not be concerned with units.

$($a$)$ Calculate the $y-$component of velocity, $v.$

$($b$)$ Regardless of your answer to Part $($a$),$ set $v=5y+2,$ and calculate the stream function $(x,y).$

$($c$)$ Generate an expression for the pressure field associated with this velocity field, i$.$e$.,$ calculate $P(x,y).$ Is this a valid steady, two$-$dimensional, incompressible, laminar flow field? Explain. Hint: To calculate the pressure field, you will need to write out both the $x$ and $y$ components of the Navier$-$Stokes equation, reduce them as much as possible, and then integrate them to calculate $P(x,y),$ noting that the pressure derivatives are partial derivatives, so the integration must be handled carefully, as discussed in class.