Answer
The given equation has symmetry about the x-axis only.
Work Step by Step
Step I: To check symmetry about the y-axis:
Putting $x=-x$ in the given equation, if the equation remains the same, then it has symmetry about the y-axis.
$\begin{align}
& {{x}^{3}}-{{y}^{2}}=5 \\
& {{\left( -x \right)}^{3}}-{{y}^{2}}=5 \\
& -{{x}^{3}}-{{y}^{2}}=5
\end{align}$
It is not the same as the provided equation, hence it is not symmetric about the y-axis.
Step II: To check symmetry about the x-axis:
Putting $y=-y$ in the given equation, if the equation remains the same, then it has symmetry about the x-axis.
$\begin{align}
& {{x}^{3}}-{{y}^{2}}=5 \\
& {{x}^{3}}-{{\left( -y \right)}^{2}}=5 \\
& {{x}^{3}}-{{y}^{2}}=5
\end{align}$
It is the same as the provided equation, hence it has symmetry about the x-axis.
Step III: To check symmetry about the origin:
Putting $x=-x\text{ and }y=-y$ in the given equation, if the equation remains the same, then it has symmetry about the origin.
$\begin{align}
& {{x}^{3}}-{{y}^{2}}=5 \\
& {{\left( -x \right)}^{3}}-{{\left( -y \right)}^{2}}=5 \\
& -{{x}^{3}}-{{y}^{2}}=5
\end{align}$
It is not the same as the provided equation, hence it is not symmetric about the origin.
Therefore, the given equation has symmetry about the x-axis only.
