5. The state-space representation for 2x" + 4x + 5x = 10e" is D = +² го 98
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![5. The state-space representation for 2x" +4x' + 5x = 10e is
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- Prove statement d of Theorem 3.9: If G is abelian, (xy)n=xnyn for all integers n.Compute the Wronskian for 2t^2y''+3ty'-y=0What are the eigenvalues and eigenfunctions: x′′ + λx = 0, x(1) = x(3), x′(1) = x′(3) - Definey by x(z) = y((z−2)π) for 1 ≤ z ≤ 3. Show that y(−π) = y(π) and y′(−π) = y′(π) - Substitute into the equation to get y′′((z−2)π) + (λ/π2) y((z−2)π), for 1 ≤ z ≤ 3 - Use the change of variable t = (z − 2)π to show that the above equation has a non-zero solution if and only if either λ = k2π2 for some integer k ≥ 1 or λ = 0 and the solutions (eigenfunctions) are given by cos(kt), sin(kt) and 1 for −π ≤ t ≤ π. - Plug back t = (z − 2)π to find the eigenfunctions and eigenvalues of original equation
- 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.For what values of x can the eigenvalue problem:(a) Find a conjugacy C between G(x) = 4x(1-x) and g(x)=2-x^2 . (b) Show that g(x) has chaotic orbits.
- Consider the ODE eigenvalue problem: ((1-x2)1/2φ')' + λ(1-x2)-1/2φ = 0 posed for -1 < x < 1 and subject to the boundary condidiont |φ(-1)| < ∞ and |φ(1)| < ∞. If you find it helpful you, you may assume |φ'(-1)| < ∞ and |φ'(1)| < ∞. a) Show that this is a Sturm-Liouville eigenvalue problem, i.e, verify that it has the correct form and identify the coefficients p, q, and σ. Is the problem regular? Why or why not? b) Show that λ > 0 for each eigenvalue λ.Let f(x) = xT Ax be a quadratic form with associated n X n symmetric matrix A. Let the eigenvalues of A be λ1 >= λ2>= ···>= λn Then the following are true, subject to the constraint II x|| = 1: Prove The minimum value of f(x) is λn, and it occurs when x is a unit eigenvector corresponding to λnConsider the following system of equations over the finite field Z3 x + 2y + z = 1x + z = 1x + y + z = 1 (a) What is the reduced row echelon form of the associated augmented matrix? Write down the sequence of operations you performed to obtain the reduced row echelon form. (b) Describe the solution set and state how many different solutions are there.