Methods | Part IB, 2002

A function y(x)y(x) is chosen to make the integral

I=abf(x,y,y,y)dxI=\int_{a}^{b} f\left(x, y, y^{\prime}, y^{\prime \prime}\right) d x

stationary, subject to given values of y(a),y(a),y(b)y(a), y^{\prime}(a), y(b) and y(b)y^{\prime}(b). Derive an analogue of the Euler-Lagrange equation for y(x)y(x).

Solve this equation for the case where

f=x4y2+4y2yf=x^{4} y^{\prime \prime 2}+4 y^{2} y^{\prime}

in the interval [0,1][0,1] and

x2y(x)0,xy(x)1x^{2} y(x) \rightarrow 0, \quad x y(x) \rightarrow 1

as x0x \rightarrow 0, whilst

y(1)=2,y(1)=0y(1)=2, \quad y^{\prime}(1)=0

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