Question

Solve Utt = c² Uxx​,c > 0 x∈ IR I.c. {
u (t = 0,x) = sinx
ut (t=0,x)=0.
​

Answer 1

The solution to the equation Utt = c²Uxx, with given initial conditions, is u(t, x) = sin(x + ct), where u is a sinusoidal wave moving to the right with speed c.

Step-by-step explanation:

To solve the equation Utt = c²Uxx, where c > 0 and x ∈ IR, we need to use the given initial conditions. The initial condition u(t = 0, x) = sin(x) indicates that the function u is sinusoidal in nature. Since ut(t = 0, x) = 0, it means that the function u is at rest initially and has no initial velocity. Applying the given equation and initial conditions, the solution for u(t, x) is u(t, x) = sin(x + ct). This represents a sinusoidal wave moving to the right with speed c.