Question

The function u(z, y) satisfies the partial differential equation \frac{\partial u}{\partial x}-y \frac{\partial u}{\partial y}=-2 u x subject to u(0, \tau)=\tau \quad \text { for } \quad|\tau|<1 Write down the

equations for the characteristics in the z-y plane. Show that the characteristic that passes through the point (z. y) = (0.T) is given by y=Te. Sketch the characteristics on a graph of the z-y plane. Show the domain of definition on your sketch, and describe it using one or more inequalities involving z and/or y. Obtain an expression for du/dz along a characteristic, and solve this equation to obtain anexpression for u along the characteristic that passes through (0, 7). Hence find the solution foru(z, y) within the domain of definition.

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