Verify by mathematical induction 1. 2(2i– 1)° = n²(2n² – 1) for all n21. 33 n(n+1) 2. Prove: Vn e Z*, i = 2
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- Use the method of eigenfunction expansions to solve the following heat ow problemwith external heat source in a one-dimensional rod:@u@t= 2@2u@x2 + e?(x+t) 0 < x < ; t > 0;u(0; t) = 0;u(; t) = 0;u(x; 0) = ?x:You can use the provided formula sheets.Pagea. Show that W[5, sin2 t, cos(2t)] = 0 for all t by directly evaluating the Wrosnkian. b. Establish the same result without direct evaluation of the Wronskian.Q2. (a) Show that \{e ^ (- x), x * e ^ (- x)\} is a fundamental set of solutions of y^ prime prime +2y^ prime +y=0 on(0, ∞)
- Solve the IVP, utt −uxx =sin(3πx)−7sin(4πx),0≤x≤1t∈R+ IC u(x,0) = 5/4 sin(6 pi x) +1/3sin(4pi x) ut (x,0) = 0with the BC u(0,t)=0=u(1,t),t∈R + Express the solution as a Fourier series.Find the second order Taylor expansion of [15] f(x,y) = (1 +4x2 +y2)1/2 about the point (1,2) and use it to compute approximately (1.2,2.5).Consider an oscillator satisfying the initial valueproblem u''+w2u=0, u(0)=u0, u'(0)=v0 (i) (a)let x1=u, x2=u', and transformequation (i) into the form: x'=Ax, x(0)=x0 (ii) (b)By using the series (23) on page 417 which is (exp(At)=I + Σ∞n=1 (Antn/n!) ), show that expAt=I cos wt +A (sinwt)/w (iii) (c)Find the solution of the initial value problem (ii)
- Consider the operator d^2/dx^2. If the eigen function for above operator as tan(nx) where n=1,2,3,...... Compute the eigen value for the given eigen function.1. Find the natural cubic spline sN (x) passing through the 3 points (xj, yj) given by (0, 2), (2, 3), and (3, 1).Then evaluate sN (1).What is the eigenvalue for the eigenfunction e^4ix of the operator ((d^2)/(dx^2))+(d/dx)−4i?
- Consider an oscillator satisfying the initial value problem (50) u′′+ω2u=0,u(0)=u0,u′(0)=v0 a.Let x1 = u, x2 = u′, and transform equations (53) into the form (51) x′=Ax,x(0)=x0. b.Use the series (23) to show that (52) exp(At)=I cos(ωt)+Asin(ωt)ω.true or false In the domain of Gaussian integers Z[i], the element 17 is Irreducible5. For f(x) = 1 –x2on [−2, 1], do the hypotheses and conclusion of Rolle’s Theorem hold?