Answer
The wave equation is satisfied.
Work Step by Step
We need to compute the Wave equation.
The wave equation can be defined as: $c^2 \dfrac{\partial^2 w}{\partial x^2}= \dfrac{\partial^2 w}{\partial t^2}$
In order to find the partial derivative, we will differentiate $w$ with respect to $t$, by keeping $x$ and $c$ as a constant, and vice versa:
$\dfrac{\partial w}{\partial t}=\dfrac{\partial (5 \cos (3x+3ct))}{\partial t}+\dfrac{\partial (e^{x+ct})}{\partial t}=-15c \sin (3x+3ct) +ce^{x+ct}$ and $\dfrac{\partial w}{\partial x}= -15 \sin (3x+3ct) +e^{x+ct}$
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}...(1)$
and $c^2 \dfrac{\partial^2 w}{\partial x^2}=c^2 \dfrac{\partial}{\partial x}( -15 \sin (3x+3ct) +e^{x+ct})=-45c^2 \cos (3x+3ct) +c^2e^{x+ct} ......(2)$
Thus, equations (1) and (2) are equal, so the wave equation is satisfied.
