Paper 2, Section II, A

Differential Equations | Part IA, 2016

(a) For each non-negative integer nn and positive constant λ\lambda, let

In(λ)=0xneλxdxI_{n}(\lambda)=\int_{0}^{\infty} x^{n} e^{-\lambda x} d x

By differentiating InI_{n} with respect to λ\lambda, find its value in terms of nn and λ\lambda.

(b) By making the change of variables x=u+v,y=uvx=u+v, y=u-v, transform the differential equation

2fxy=1\frac{\partial^{2} f}{\partial x \partial y}=1

into a differential equation for gg, where g(u,v)=f(x,y)g(u, v)=f(x, y).

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