# pde – Solving the wave equation with initial conditions \$ u_x = 0 \$ in \$ x = 0, pi \$

So, we have,

$$u_ {tt} -c ^ 2u_ {xx} = 0$$, with initial conditions $$u_x = 0$$ to $$x = 0, pi$$

Without loss of generality, let's take $$c = 1$$, and suppose that our solution is in the form. $$u (x, t) = X (x) T (t)$$. Then we can rewrite our equation as,

$$XT ^ {& # 39; & # 39;} = X ^ {& # 39; & # 39;} T$$

$$frac {T ^ {& # 39; & # 39;}} {T} = frac {X {& # 39; & # 39;}} {X} = – lambda$$, where $$lambda> 0$$ It is our separation constant.

Solving the X equation:

$$X ^ {& # 39; & # 39;} + lambda X = 0 implies X = Acos ( sqrt { lambda} x) + Bsin ( sqrt { lambda} x)$$

We differentiate this equation to apply the initial conditions.

$$X ^ {& # 39;} = – sqrt { lambda} A sin ( sqrt { lambda} x) + sqrt { lambda} B cos ( sqrt { lambda} x)$$

Using the first initial condition we obtain:

$$0 = sqrt { lambda} B implies B = 0$$

Then our $$X ^ {& # 39;}$$ the equation is reduced to

$$X ^ {& # 39;} = – sqrt { lambda} A sin ( sqrt { lambda} x)$$

Applying the second initial condition you get …

$$0 = – sqrt { lambda} A sin ( sqrt { lambda} pi)$$, for a non-trivial solution we solve $$sin ( sqrt { lambda} pi) = 0$$

$$implies sqrt { lambda} pi = n pi implies sqrt { lambda} = n$$

So the equation of X is,

$$X ^ {& # 39;} = -nA_n sin (n pi x)$$

Then integrating,

$$X = frac {A_n} { pi} cos (n pi x)$$

Is this the correct solution for the X equation? Thank you!