Independent and Dependent Variable Meaning? Given a differential equation, for example: 

$\displaystyle \frac{d^2x}{dt^2} + a \frac{dx}{dt} + kx = 0$

Is there a reason why we call $x$ the dependent variable and $t$ the independent variable? Here is the definition from my textbook: 

If an equation involves the derivative of one variable with respect to another, then the former is called a dependent variable and  the latter an independent variable

So according to this definition it makes sense why $x$ and $t$ are assigned in this way but is there an intuitive reasoning behind this terminology? 

As others have said, $x$ is a function of $t$. Differential equations like these were normally studied in the context of science, so it would describe physical phenomena. One might want to measure "how does the pressure of a gas change when I vary its temperature?" To do that, you would vary your independent variable (temperature) and observe changes in the dependent variable (pressure). Pressure depends on temperature. This experiment gives you a function $P(T)$. The vocabulary for experiments carried on into functions.
