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Numerade Educator

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Problem 14 Medium Difficulty

A curve passes through the point (0, 5) and has the property that the slope of the curve at every point $ P $ is twice the $ y- $ coordinate of $ P. $ What is the equation of the curve?

Answer

$y=5 e^{2 x}$

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Video Transcript

in this problem. We're told that we have a curve whose slope is equal to twice the y coordinate. And you can think of slope as the derivative D Y D X, and then twice the y coordinate would be two times why. So when we see that equation, we might think about the main idea from this section where if you have a differential equation of the form, do I d t equals K times. Why it's solution will be this exponential growth and decay formula. So what that tells us is that we have something whose solution will be, Why equals? Why not times e to the K t. And in this case, que is too. And we're using X instead of t. Okay, so now what we need to know is the value of why not and that will complete are figuring out the model or the equation. So let's use the ordered pair that was given, we know goes through the 0.5 so we can substitute five and for the y value and zero in for the X value. And we have five equals. Why not times e to the two time zero So that would be five equals. Why not tons? Each of the zero each of the zero is one so five equals. Why not times one. So now we realize that why not? Is five. So if you substitute that in, you have the equation y equals five times e to the two x.