Partial Differential Equations 1

Calculus Level 4

Solve \[\frac{{\partial}^{2} z}{\partial x \partial y} = {x}^{2} y\] given the conditions \(z(x,0)={x}^{2}\) and \(z(1,y) = \cos {y} \).

The solution is of the form \[z(x,y) = A{x}^{3} {y}^{2} + B \cos {y} - C{y}^{2} +D{x}^{2} -1.\] What is \(A + B + C + D\)?

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