Question

What is a solution to the differential equation `dy/dx=y`?

Answer 1

`y = C*e^x` where `C` is some constant.

Explanation:

If you aren't looking for the general solution, but rather just one solution, then sometimes you can figure it out for simple differential equations like this by thinking for a second about what the differential equation literally means.

`dy/dx=y`

We're looking for a function, `y`, which has the property that the derivative of `y` is equal to `y` itself.

There's one function which you probably learned previously that has exactly this property:

`y = e^x`.

The function `e^x` is so special precisely because its derivative is also equal to `e^x`. So `y = e^x` is one solution to the differential equation.

If you're also interested in finding all solutions to this DE, (or you're not interested in trial-and-error) then you can solve this DE by separation of variables.

Think of `dy` and `dx` each as discrete variables. So you could do something like multiply both sides by `dx` and end up with:

`iff dy=ydx`

And then divide both sides by `y`:

`iff dy/y=dx`

Now, integrate the left-hand side `dy` and the right-hand side `dx`:

`iff int 1/y dy=int dx`

`iff ln |y|=x+C`

Remember to add the constant of integration, but we only need one.

Raise both sides by `e` to cancel the `ln`:

`iff y=+-e^(x+C)`

Now, pulling the `C` out front:

`iff y=+-Ce^x`

Since `C` can be either positive or negative, we don't really need the `+-`:

`iff y=Ce^x`

So there is our general solution: Any constant multiple of `e^x` is a solution to the differential equation, which makes sense.