Paper 2, Section II, B

Differential Equations | Part IA, 2015

Suppose that u(x)u(x) satisfies the equation

d2udx2f(x)u=0\frac{d^{2} u}{d x^{2}}-f(x) u=0

where f(x)f(x) is a given non-zero function. Show that under the change of coordinates x=x(t)x=x(t),

d2udt2x¨x˙dudtx˙2f(x)u=0\frac{d^{2} u}{d t^{2}}-\frac{\ddot{x}}{\dot{x}} \frac{d u}{d t}-\dot{x}^{2} f(x) u=0

where a dot denotes differentiation with respect to tt. Furthermore, show that the function

U(t)=x˙12u(x)U(t)=\dot{x}^{-\frac{1}{2}} u(x)


d2Udt2[x˙2f(x)+x˙12(x¨x˙ddt(x˙12)d2dt2(x˙12))]U=0\frac{d^{2} U}{d t^{2}}-\left[\dot{x}^{2} f(x)+\dot{x}^{-\frac{1}{2}}\left(\frac{\ddot{x}}{\dot{x}} \frac{d}{d t}\left(\dot{x}^{\frac{1}{2}}\right)-\frac{d^{2}}{d t^{2}}\left(\dot{x}^{\frac{1}{2}}\right)\right)\right] U=0

Choosing x˙=(f(x))12\dot{x}=(f(x))^{-\frac{1}{2}}, deduce that

d2Udt2(1+F(t))U=0\frac{d^{2} U}{d t^{2}}-(1+F(t)) U=0

for some appropriate function F(t)F(t). Assuming that FF may be neglected, deduce that u(x)u(x) can be approximated by

u(x)A(x)(c+eG(x)+ceG(x)),u(x) \approx A(x)\left(c_{+} e^{G(x)}+c_{-} e^{-G(x)}\right),

where c+,cc_{+}, c_{-}are constants and A,GA, G are functions that you should determine in terms of f(x)f(x).

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