Consider the function Let where f(t) is differentiable for all t ∈ R. Show that z satisfies the partial differential equation (x2 − y2 ) ∂z/∂x + xy ∂z/∂y = xyz for all (x, y) ∈ R2 \ { (t, 0)|t ∈ R }.
soi-Ja x(rprr) a r, where x(r) is continuous at t-o.anda <0< β. 3.13 Show that (a) (t - T)s-T)0, (c) cos(1)s(t + π/2),: 0, 3.14 Evaluate the following definite integrals: (a) sin(r)s)dr, (b) o sinoo)dt (c) sin(r)8(r)a(t-2) dr, τ cos(r/2)δ(r-x) dr. soi-Ja x(rprr) a r, where x(r) is continuous at t-o.anda
i need solution for this question 7. The zeroth-order perturbation in Question 6 satisfies with Uo(X,0 0,U0(T, T)-1, and Uo(oo, T) 0 (a) Using change of variable, ξ-X-T, τ-T, show that the governing equation becomeand express the initial and boundary conditions in terms of the new variables. (b) Obtain a quasi-steady -state solution by neglecting the derivative with respect to t. Express this back in terms of the variables X and T. Using sketching package (or graphic calculator), explain what...
8.1. Consider the problenm min f(x) (P) t. g(x)s0 where f and g are convex functions over R" and X CR" is a convex set. Suppose that x is an optimal solution of (P) that satisfies g(x")<0. Show that x is also an optimal solution of the problem min f(x) s.t. xX. 8.1. Consider the problenm min f(x) (P) t. g(x)s0 where f and g are convex functions over R" and X CR" is a convex set. Suppose that x...
t + τ Proof From Definition 10.17, RİT (r) yields Rn(t) = Elx()r(t + τ)]. Making the substitution u Since X(0) and Y(O) are jointly wide sense stationary, Ryr(u, -t for random sequences Rx-r). The proof is similar i: 10.11X(t) is a wide sense stationary stochastic process with autocorrelation function Rx(r). (2) Express the autocorrelation function of Y(C) in terms of Rx(r) Is r) wide sense (2) Express the cross-correlation function of x(t) and Y (t) in terms of Rx(t)...
Bayesian updating Suppose that we have the model y|μ ~ N(μ, τ-1) where τ > 0 is known and μ is an unknown parameter (iii) Suppose that we have a prior μ ~ N(a, b-1) where b > 0, Show that the prior distribution π(A) verifies r(11) x exp (iv) Show that the posterior π(μ|y) verifies (v) which distribution is π(μ|y)? Bayesian updating Suppose that we have the model y|μ ~ N(μ, τ-1) where τ > 0 is known and...
x(t)-A()x(t) + B(t)u(t) 2. Consider the following system y(t)-C(t)x(t) where A(t)- 0 1) Derive the state transition matrix Φ(t, τ). 2) Derive the impulse response function g(t, z). x(t)-A()x(t) + B(t)u(t) 2. Consider the following system y(t)-C(t)x(t) where A(t)- 0 1) Derive the state transition matrix Φ(t, τ). 2) Derive the impulse response function g(t, z).
(a) Show that dB/ds is perpendicular to B 0 BI= 1-B-B- (b) Show that dB/ds is parpendicular to T B Tx N T' N' [(T'x N)(T * N Irtt) rit) [(TTT (Tx N rt) (c) Doduce from parts (a) and (b) that dB/ds -risN for some numbar ria called the torsion of the curve, (The torsion meacures the dearoe of twisting of a curve.) TIN. P LT and BI N. So B T and N torm -Select set of vectors...
P(x,t) = Aeixe-ißt a) Show that the above function is a wave by showing that it satisfies the wave equation. A, a, B are arbitrary constants, i is the unit imaginary number. b) Find the wave speed where a = 1, B = 4, and A-3.
Change variables under the Laplace integral to prove that for a fixed τ, L{x(t−τ)}=e^(−sτ) X(s)