Paper 2, Section II, B

Differential Equations | Part IA, 2015

Consider the equation

22ux2+32uy272uxy=02 \frac{\partial^{2} u}{\partial x^{2}}+3 \frac{\partial^{2} u}{\partial y^{2}}-7 \frac{\partial^{2} u}{\partial x \partial y}=0

for the function u(x,y)u(x, y), where xx and yy are real variables. By using the change of variables

ξ=x+αy,η=βx+y\xi=x+\alpha y, \quad \eta=\beta x+y

where α\alpha and β\beta are appropriately chosen integers, transform ()(*) into the equation

2uξη=0\frac{\partial^{2} u}{\partial \xi \partial \eta}=0

Hence, solve equation ()(*) supplemented with the boundary conditions

u(0,y)=4y2,u(2y,y)=0, for all yu(0, y)=4 y^{2}, \quad u(-2 y, y)=0, \quad \text { for all } y

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