. Consider the following problem for Laplaceís equation in a rectangle. @ 2u @x2 (x; y) + @ 2u @y2 (x; y) = 0 for all (x; y) in the rectangle [0; 8] [0; 4] which is all (x; y) where 0 x 8 and 0 y 4 and u(x; y) = F(x; y) for all (x; y) on the boundary of [0; 8] [0; 4] where F(x; y) = x 3 + y 2 + xy + 2 Find the function v of the form v(x; y) = ax + by + cxy + d such that v(x; y) = F(x; y

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step-by-step explanation:

we have three similar triangles, because each has a right angle and shares an angle.   let's write the angles in order: opposite to short leg, long leg, hypotenuse.

cab similar to bad similar to cbd

or as ratios,

ca: ab: cb = ba: ad: bd = cb: bd: cd

we also know

ac = ad + cd

(ad+cd): ab: cb = ba: ad: bd = cb: bd: cd

(ad+cd)/cb=cb/cd

we have cb=6, ad=5 and seek x=cd.

we reject the negative root and conclude x=4

answer: 4