0 Find the explicit finite difference solution of the wave equation utt -uxx =0, 00, along with the following initial and boundary conditions: u(x,0)=sin(x), ut(x,0) = 0, 0 ≤ x ≤ 1, u(0,t)=0, u(1,t)=0, t≥0 By Laplace transform

Solve the wave equation for the following cases: ii) For free free ends beam with following condition: du Boundry conditions: For x = 0 and x = 1 => əx Inetial conditions: inetial desplacement: u(x, 0) = f(x): du(x, 0) inetial velocity at = g(x). ||

Show directly that the trial wave function (x, t) = e(kx-wt) satisfies in- The derivatives are (Use the following as necessary: Y, k, and w.) i(kx-cot) and 24 at 2² əx² hwy = -iwe The result is -kei(kx-cot) 2 -y + Vy 2m X X a4(x, t) at Substituting in the time-dependent Schrödinger equation gives (Use the following as necessary: Y, p and m.) + VY = ħ² a²(x, t) 2m əx² + VY(x, t). E = K + V where K is the kinetic energy, which is a statement of conservation of mechanical energy.

solve the folloing : use properities and the table of basic functions 3s +25 1) c- 5) y=C-1 s+6s+13 2) c-- (s - 3)5 6) Use Laplace transforms to solve the differential equation: dfy dy -7 +10y=c+20, given that when x-0, y=0 dx 75 3) si+4 dr? dy and dx 1 4) C-

A string of length 'L', that has been tightly stretched between two points at X-axis for t > 0, and satisfied one-dimensional wave equation of the following form: 8²u = 4. St2 82u 8x2 Produce the wave function, u(x,t) using separation of variables method with given boundary and initial conditions below. u(0, )- и (п,1) -0, u(x,0)= sin 5x t > 0 ди (at t=0) = 0 %3D

Show that cos(ωt − β), cos ωt, sin ωt are linearly dependent functions of t.

3 The centre point of a perfectly elastic string stretched between two points A and B, 4 m apart, is deflected a distance 0.01 m from its position of rest perpendicular to AB and released initially with zero where c= 10 to 2²u 1 2u Əx² c² at² velocity. Apply the wave equation determine the subsequent motion of a point P distant x from A at time t. =

A string with motionless ends at x = 0 and x = 1 vibrates according to the wave equation Fu Ət² and the initial velocity J²u dr² 1. Use separation of variables (show details) to solve the equation provided that the initial profile of the string is u(x,0) = 4 sin(2x) Ju 5- Ət It=0 = 0.

a2u satisfies the wave equation əx² -n a²u Verify that U(x, t) = e¬Vkt cos\ax %3D k at2

for wave equation, seperation of vairables u(x,t)=X=(x)T(t)