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

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Oct 13, 2016

Answer:

#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.

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