4) Solve Poisson problem for a=lı b=l ↑u (x₁b) = f₂(x) ty b (0₁4)=9₁(y) 0 Ou=f(x,y) u(a.y) = 9₂ (9) xx a u(x₁) = f(x) Figure 1. f(x,y) = sin2TX₁ f₁= £₂=0, 9₁=9₂=0 using one dimensional eigen function expansion. 1

In implicit differentation problems it seems we treat x and y differently. For example, when we differentiate both sides of cos(x) + sin(y) = x^3 + y^3, we get -sin(x) + cos(y)y' = 3x^2 + 3y^2y'. a) Why do we get y' but but not a x'? WHy are these variables treated differently? b) How is this related to the chain rule?

(7) (y sec 2x + sec x tan x)dx + (tan x + 2y)dy = 0 (8) (22) ds + (금 () dt = 0 t (9) (2x³ – 3x²y + y³) = 2x³ – 6x²y + 3xy? dx