Problem (F'90, #8). Consider the differential equation -4.1+(1+x2)ur du, u(0) = u(a)=0. a) Find a variational characteri

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Problem F 90 8 Consider The Differential Equation 4 1 1 X2 Ur Du U 0 U A 0 A Find A Variational Characteri 1 (32 KiB) Viewed 53 times

Problem (F'90, #8). Consider the differential equation -4.1+(1+x2)ur du, u(0) = u(a)=0. a) Find a variational characterization for the eigenvalues li, i = 1, 2, .... b) Show that the eigenvalues are all positive, i.e. l;> 0. c) Consider the problem for two different values of a: a = a1 and a = a2 with aj < 22. Show that the eigenvalues 1(a) for a = ay is larger than or equal to the first eigenvalues 11(02) for 12, i.e. 41(21) 2 11(az) for ai < 02. d) Is this still true for i > 1, i.e. is A(Q1) (a) for a <a ? 0.