Show that the function is a solution of the wave equation.
-w(x, t) = cos ( ct) sin ( x)

Question

Quizplus · Accepted Answer

The wave equation is given by: 
$$\frac{\partial^2 w}{\partial t^2} = c^2\frac{\partial^2 w}{\partial x^2} $$
Taking the partial derivatives of the given function with respect to time and distance, we get:
$$\frac{\partial w}{\partial t} = -c\sin(ct)\sin(x)$$ 
$$\frac{\partial^2 w}{\partial t^2} = -c^2\cos(ct)\sin(x)$$ 
$$\frac{\partial w}{\partial x} = \cos(ct)\cos(x)$$ 
$$\frac{\partial^2 w}{\partial x^2} = -\cos(ct)\sin(x)$$ 
Substituting these derivative values into the wave equation, we get:
$$-c^2\cos(ct)\sin(x) = c^2(-\cos(ct)\sin(x))$$ 
Simplifying, we get:
$$-c^2\cos(ct)\sin(x) = -c^2\cos(ct)\sin(x)$$ 
Thus, the given function w(x,t) satisfies the wave equation, and the answer is True (A).

Show That the Function Is a Solution of the Wave