Consider the following problem:

*y*' = 5*y*

This is a first-order linear ordinary differential equation in which the derivative of *y* with respect to *x* (denoted *y*’) is 5 times the value of *y* itself at any given point. Note that *y*’ is the same as *dy */ *dx*. Now observe:

We began by dividing by y to isolate the 5. We then multiplied by *dx*, at which point we integrated both sides. We got right of the ln term on the left by exponentiating, and we relied on the properties of exponents to simplify further. Noting that *e*^{C} is a constant, we followed the convention of renaming it as K. Then, to remove the absolute value sign around y, we used a plus or minus sign on the right.

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