Answer
The wave equation is satisfied.
Work Step by Step
We need to prove the wave equation $c^2 \dfrac{\partial^2 w}{\partial x^2}= \dfrac{\partial^2 w}{\partial t^2}$
We take the first partial derivative of the given function $w(t,x)$ with respect to $x$, by treating $t$ as a constant, and vice versa:
$\dfrac{\partial w}{\partial t}=-15c \sin (3x+3ct) +ce^{x+ct} \\ \dfrac{\partial w}{\partial x}= -15 \sin (3x+3ct) +e^{x+ct}$
Now, $ \dfrac{\partial^2 w}{\partial t^2}= \dfrac{\partial}{\partial t}( -15c \sin (3x+3ct) +ce^{x+ct})= -45c^2 \cos (3x+3ct) +c^2e^{x+ct}$
and $c^2 \dfrac{\partial^2 w}{\partial x^2}=c^2 \dfrac{\partial}{\partial x}( -15 \sin (3x+3ct) +e^{x+ct})$
or, $=-45c^2 \cos (3x+3ct) +c^2e^{x+ct}$
So, $c^2 \dfrac{\partial^2 w}{\partial x^2}= \dfrac{\partial^2 w}{\partial t^2}$
Thus, it has been proved that the wave equation is satisfied.
