I am being asked to write the non-dimensional version of the equation $ae^{bx}+c=x^2$. I understand the process that one would use to non-dimensionalise an equation, but in this case it was the exponential that stumped me - I am not sure whether I have done it correctly.
I am given that $[x]=kg$. Using the fact that all equations are dimensionally homogeneous, I have deduced that $[a]=kg^2$, $[b]=kg^{-1}$ and $c=kg^2$.
I can write $x=Xx'$, where $X$ is a dimensional constant and $x'$ is a dimensionless variable. Substituting this back into the original equation gives $ae^{bXx'}+c=X^2x'^2$.
Dividing the equation by $X^2$ gives $\alpha e^{bXx'}+\beta =x'^2$, where $\alpha =\frac{a}{X^2}$ and $\beta = \frac{c}{X^2}$.
I am uncertain about what to do next, as there is still a dimensional term $X$ in the exponent. Or have I finished?
The dimension of $X$ cancels out the dimension of $b$ so you are done. Similar to how $\beta$ and $\alpha$ are dimensionless, you can call $b X$, perhaps, $\gamma$ and it would also be dimensionless.