Separate the following partial differential equation into a set of three ODEs by the method of separation of variables:
d^2u/dt^2 =c^2,[d^2u/dr^2+1/r du/dr+1/r^2 d^2u/d theeta^2]
To separate the given partial differential equation (PDE) into a set of three ordinary differential equations (ODEs) using the method of separation of variables, we need to express the PDE in terms of three separate variables: t, r, and θ.
Let's first focus on the t variable. The PDE provided is:
d^2u/dt^2 = c^2
To separate this equation into an ODE, we assume that the solution is of the form: u(t, r, θ) = T(t)R(r)Θ(θ).
Now, let's substitute the separation of variables form into the PDE:
T''(t)R(r)Θ(θ) = c^2
Divide both sides of the equation by T(t)R(r)Θ(θ) to separate variables:
T''(t)/T(t) = c^2/(R(r)Θ(θ))
As the left-hand side of the equation depends only on t and the right-hand side depends only on r and θ, both sides must be equal to a constant to satisfy the equation. Let's call this constant -λ^2:
T''(t)/T(t) = -λ^2 = c^2/(R(r)Θ(θ))
Now, we have two separate equations:
1) T''(t)/T(t) = -λ^2
2) c^2/(R(r)Θ(θ)) = -λ^2
Let's work on the second equation involving r and θ.
The given equation involving r and θ is:
(d^2u/dr^2) + (1/r)(du/dr) + (1/r^2)(d^2u/dθ^2)
Substitute u(t, r, θ) = T(t)R(r)Θ(θ) into the equation above:
(R''(r) + (1/r)R'(r) + (1/r^2)R(r))Θ(θ) = -λ^2c^2
Divide both sides by R(r)Θ(θ) to separate variables:
(R''(r) + (1/r)R'(r) + (1/r^2)R(r)) = -λ^2c^2
Now, we have two more separate equations:
2) c^2/(R(r)Θ(θ)) = -λ^2
3) (R''(r) + (1/r)R'(r) + (1/r^2)R(r)) = -λ^2c^2
To complete the separation of variables, we need to solve each of these equations separately. Equation 1) gives us T(t), equation 2) gives us Θ(θ), and equation 3) gives us R(r). These three solutions will form a set of three ODEs that arise from the separation of variables of the given PDE.