Verify that the solution of the Rodrigues formula for l = 2 is also a solution of the Laplace equation in spherical coordinate (assuming x = cos θ)
Verify that the solution of the Rodrigues formula for l = 2 is also a solution...
PDE Greens function: 2. In class we constructed the Green's function for the Laplace operator on the disc with Dirichlet boundary conditions and found that G is given by G(x.xo)-. In (K-xo)-1 In (빻) CU where xXo xol2 Use this Green's function to construct the solution of the equation u(a, θ) = g(θ) and verify the Poisson integral formula (r- |x|) 2π C0 r" 0. ar coS 2. In class we constructed the Green's function for the Laplace operator on...
Verify the following using MATLAB 2) (a) Consider the following function f(t)=e"" sinwt u (t (1) .... Write the formula for Laplace transform. L[f)]=F(6) F(6))e"d Where f(t is the function in time domain. F(s) is the function in frequency domain Apply Laplace transform to equation 1. Le sin cot u()]F(s) Consider, f() sin wtu(t). From the frequency shifting theorem, L(e"f()F(s+a) (2) Apply Laplace transform to f(t). F,(s)sin ot u (t)e" "dt Define the step function, u(t u(t)= 1 fort >0...
PLE 2 The point (0, 5 3 , −5) is given in rectangular coordinates. Find spherical coordinates for this point. SOLUTION From the distance formula we have ρ = x2 + y2 + z2 = 0 + 75 + 25 = 10 Correct: Your answer is correct. and so these equations give the following. cos(φ) = z ρ = -1/2 Correct: Your answer is correct. φ = $$ Incorrect: Your answer is incorrect. cos(θ) = x ρ sin(φ) = θ...
Verify that the indicated function is an explicit solution of the given differential equation. Give an interval of definition I for the solution. y" + y = sec(x); y = x sin(x) + (cos(x)) In(cos(x)) O [0,7) O (-0,0) O (-0,-) O (0 ) O(
verify the following trigonometric identities. cos y 1-sın y 5, sec y + tany= cos x-sin x -cosx 1-tanx sinx cosx-l 7. sin20+cos 2 θ+ cot 2a 1+tan 2 θ 8.
Problem 1 Use residues to verify the formula poo cos(3x) – cos(x) dx = a Jo x² Hint: use the indented contour with two semicircles from the April 14 lecture. Problem 2 Use residues to verify the formula [ * nedz =
Empty Part only Let L[y]: y"" y'+4xy, yi (x): = sinx, y2(x): =x. Verify that L[y11(x) 4xsinx and to the following differential equations. Ly2 (X)= 4x1. Then use the superposition principle (linearity) to find a solution (a) Lly] 8x sin x - 4x2-1 (b) Lly] 16x+4 -24x sin x y1(x)- cos x tlV]¢»= 4x° Substituting yi (x), y, '(x), and y"(x) into L[y] y""+y' +4xy yields Lfy1(x) 4xsinx. Now verify that +1. Calculate y2'(x) y2'(x) 1 Calculate y2"(x). У2"(х)%3D 0...
8 and 9 IQ 8. Verify that the function =- √2² + y² + 2 is a solution of the three dimensional Laplace equation Uzz + Uyy + x = 0. 9. Consider the general homogeneous linear second order partial differential equal tion Aunu +2Buay + Cuw + Dus + Euy + Fu =0. (a) Show that the function u(x, y) = euseby is a solution if and only if a and b satisfy the equation Aa? + 2 Bab...
. 1. The relations for the potential, electric and magnetic fields, and time averaged intensity are written in terms of spherical coordinates (cos θ pow ___ (_) sin[w(t-r/c)] V(r, θ, t) Scalar Potential: Hopow ATT -pop Α(r, θ,t) -sin[w(t-r/c)]2 Vector Potential: μοΡου-(-) cos[w(t-r/c)ja Electric Field: Ε--7V (3) 4π 7" Magnetic Field: B Vx A- 〈S) = (ww) sn-2- Intensity: Express these relationships in "coordinate-free" form, in which one is not committed to the spherical coordinate system. As an example, Po...
verify that the given function is a solution to the given differential equation (c1 andc2 arbitrary constants), and state the maximum interval over which the solution is valid. 8. y(x) = cj cos 2x + c2 sin 2x, y + 4y = 0.