Consider the 1D wave equation d^2u/dt^2 = c^2( d^2u/dx^2) with
the following boundary conditions: u(0, t) = ux (L, t) = 0 . (a)
Use separation of variables technique to calculate the eigenvalues,
eigenfunctions and general solution. (b) Now, assume L = π and c =
1. With initial conditions u(x, 0) = 0 and ut(x, 0) = 1, calculate
the solution for u(x, t). (c) With initial conditions u(x, 0) =
sin(x/2) and ut(x, 0) = 2 sin(x/2) −...
7. (a) Solve the wave equation in three dimensions for t > 0
with the
initial conditions φ(x) = A for |x| < ρ, φ(x) = 0 for |x| >
ρ, and
ψ|x| ≡ 0, where A is a constant. (This is somewhat like the
plucked
string.) (Hint: Differentiate the solution in Exercise 6(b).)
((b) Solve the wave equation in three dimensions for t > 0
with the
initial conditions φ(x) ≡ 0,ψ(x) = A for |x| < ρ, and...
Consider the differential equation x′=[2 4
-2 −2],
with x(0)=[1 1]
Solve the differential equation where x=[x(t)y(t)].
x(t)=
y(t)=
please be as clear as possible especially when solving for c1
and c2 that's the part i need help the most
Consider the following wave equation for u(t, x) with boundary
and initial conditions defined for 0 ≤ x ≤ 2 and t ≥ 0.
∂ 2u ∂t2 = 0.01 ∂ 2u ∂x2 (0 ≤ x ≤ 2, t ≥ 0) (1)
∂u ∂x(t, 0) = 0 and ∂u ∂x(t, 2) = 0 (2)
u(0, x) = f(x) = x if 0 ≤ x ≤ 1 1 if 1 ≤ x ≤ 2.
(a) Compute the coefficients a0, a1, a2, ....