6) A spherical wave is a solution of the three-dimensional wave equation of the form u(r, t), where r is the distance to the origin (the spherical coordinate). The wave equation takes the ra²u, 2 du form spherical wave equation. Change the variables v = ru to get the following PDE for v: at2 = c2 ax 7) Solve the partial differential equation by using the method of integration: = t e2t given that t = 0, x = 3e'. Explain the procedure followed in your own words. %3D

Suppose A is not diagonalizable. If the characteristic equation of A is: -λ +2^² - 2³ = 0 then dim Nul(A) = dim Nul(A - I3) =

Page 1 > of 2 ZOOM + 2. Prove the orthogonality of the sine function. Specifically, prove that (0, ifn' + n (n'пу) dy = a sin () ппу а а a if n' = = n. Hint: Y ou need to use the following relationships: 1 sin A sin B = F [cos(A - B) — сos(A + B)] sin ax lim = a x→0

Solve part B please

Question attached. Z is a complex variable, z=x+iy

Use the substitution u=-3x+2y to solvethe differential equation (-3x+2y) y'= - 1. a) Clearly classify the equation obtained after the substitution b) on the plane (x, u) sketch at least three vectors of the slope field with different slopes Attach File

2. The position vector of a particle is given by r(t)= (2 cos t sin t)i +(cos^2 t - sin^2 t)j + (3t)k If the particle begins its motion at t = 0 and ends at t = pi, find the difference between the length of the path traveled and the distance between start position and end position

Compute the quadratic form x' Ax for A = X1 5 444 X2 -9 x3 a. x = a. x¹ Ax= b. x' Ax= C. X' Ax= 1/√√3 b. x= 7 C. X = - 1/√3 1/√√3 (Simplify your answer.) 360 643 035 (Simplify your answer.) and each of the following vectors.

The two-dimensional wave equation describing the vibrations of an infinite string is 2²n_ 20²m Ət² əx² where n = n(x, t), -∞ 0 is a constant. (i) Show that the general solution can be written as n(x, t) = F(x - ct) + G(x + ct) with F and G arbitrary functions. Explain the physical interpretation of F and G in the overall shape of the string. (ii) If the string's initial position is given by n(x, t = 0) = 1 1+4x² velocity by (x, t = 0) = 0 determine F, G and hence n(x, t). Ət (iii) Now consider a semi-infinite string 0 0. If at t = = 0 the string is at rest and has displacement n(x, t 0) = x(5x)e-², find F, G and hence n(x, t), stating clearly the solution for x ct.

Find all real values of 0 such that cos 0 - sin 0 sin 0 cos O has real eigenvalues.