Paper 2, Section II, B

Differential Equations | Part IA, 2007

Find the most general solution of the equation

62ux252uxy+2uy2=16 \frac{\partial^{2} u}{\partial x^{2}}-5 \frac{\partial^{2} u}{\partial x \partial y}+\frac{\partial^{2} u}{\partial y^{2}}=1

by making the change of variables

ξ=x+2y,η=x+3y.\xi=x+2 y, \quad \eta=x+3 y .

Find the solution that satisfies u=0u=0 and u/y=x\partial u / \partial y=x when y=0y=0.

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