If u = u(x, y) = sin(y + ax) - (y + ax)^2, th | vedclass

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(A) Let $v = y + ax$. Then $u = \sin(v) - v^2$. First, we find the partial derivatives with respect to $x$: $u_x = \frac{\partial u}{\partial v} \cdot

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$u_{xx} = a^2 u_{yy}$

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(A) Let $v = y + ax$. Then $u = \sin(v) - v^2$. First, we find the partial derivatives with respect to $x$: $u_x = \frac{\partial u}{\partial v} \cdot \frac{\partial v}{\partial x} = (\cos(v) - 2v) \cdot a$. $u_{xx} = \frac{\partial}{\partial x} [a(\cos(v) - 2v)] = a(-\sin(v) - 2) \cdot \frac{\partial v}{\partial x} = a^2(-\sin(v) - 2)$. Next, we find the partial derivatives with respect to $y$: $u_y = \frac{\partial u}{\partial v} \cdot \frac{\partial v}{\partial y} = (\cos(v) - 2v) \cdot 1$. $u_{yy} = \frac{\partial}{\partial y} [\cos(v) - 2v] = (-\sin(v) - 2) \cdot \frac{\partial v}{\partial y} = -\sin(v) - 2$. Comparing $u_{xx}$ and $u_{yy}$, we see that $u_{xx} = a^2 u_{yy}$.

If $u = u(x, y) = \sin(y + ax) - (y + ax)^2$, then which of the following is true?

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