Question

Solve: u_tt = c^2 * u_xx, c>0, x in R, with initial conditions:

u(t=0, x) = sin(x)
u_t(t=0, x) = 0

Answer 1

To solve the given wave equation, we can use the method of separation of variables. By assuming the solution takes the form 'u(t, x) = T(t)X(x)', we can separate the variables and solve the resulting ordinary differential equations.

Step-by-step explanation:

The given equation is a wave equation called the one-dimensional wave equation. It describes the behavior of waves in one dimension. The equation is given by:

utt = c2uxx

To solve this equation, we need to apply the method of separation of variables. Assuming the solution takes the form:

u(t, x) = T(t)X(x)

we substitute this into the equation and separate the variables to obtain two ordinary differential equations. Solving these equations will give us the solution to the wave equation.