The differential equation is variable-separable; it can be written in the form $y\prime \left(x\right)\/y\left(x\right)\=x$ so that $f\left(y\right)\=1\/y$ and $g\left(x\right)\=x$. In this form, the equation yields immediately to antidifferentiation:

To go from the second to the third equality, exponentiate both sides: $e^{\ln \left(y\right)}\=y$ by the definition of the logarithm. On the right, be sure to exponentiate the sum. A common error is to write this as the sum of exponentials.

Note the use of the law of exponents: $e^{a\+b}\=e^a\cdot e^b$. Finally, since $c$ is arbitrary, so is $e^c$. It is generally wise to give a new name to a "new" arbitrary constant.

Finally, note that an arbitrary constant is added to just one side of the equation that results from integrating both sides of the separated differential equation. If arbitrary constants were added to both sides, the one on the left, say, could be subtracted from both sides, and on the right, the difference of two arbitrary constants would still be just one arbitrary constant.

Maple uses $\mathrm{\_C1}$ for the arbitrary constant. Names generated internally by Maple usually start with an underscore. For this reason, users are cautioned not to form names with a leading underscore.

Below, a stepwise Maple solution is implemented with commands, then interactively via the Context Panel system.

The traditional approach to solving separable differential equations makes liberal use of the differentials $\mathrm{dy}$ and $\mathrm{dx}$. Maple does not use this notation for differentials, except within typeset integrals where the differentials are written by Maple, and not by the user. Hence, the stepwise approach to solving a separable differential equation necessarily deviates from the standard textbook approach.

Notice how the integral on the left, that is, the integral of $y\prime \/y$, is in terms of the independent variable $x$. That's because the differential equation itself carries the independent variable explicitly. When the variables are separated, Maple still sees $y\left(x\right)$, not just the symbol $y$. To integrate $\mathrm{dy}\/y$ on the left would require a rewrite of the left side to just $1\/y$ and then it would be impossible to map integration across both sides of the equation - the left side is integrated with respect to $y$, but the right side is integrated with respect to $x$.

Hence, rigorously following the textbook methodology is a lot more work. If Maple is to be used to implement the solution stepwise, accommodations must be made for the structure and functioning of Maple's tools.