Consider the differential equation \frac{dy}{dx}=xy-y ,Find \frac{d^2y}{dx^2} in terms of x and...

Question:

Consider the differential equation {eq}\frac{dy}{dx}=xy-y {/eq}.

Find {eq}\displaystyle \frac{d^2y}{dx^2} {/eq} in terms of x and y.

Describe the region in the xy plane in which all solutions curves to the differential equation are concave down.

Optimization :

Derivative is the rate of change of one variable with respect to other.

Concavity Test::

If f is twice differentiable on an interval I

1. If f (x) > 0 for all x in I , f is concave upward on I.

2. If f (x) < 0 for all x I , f is concave downward on I.

Answer and Explanation:

Given ODE is

{eq}\displaystyle \frac{dy}{dx}=xy-y {/eq}

Let's differentiate w.r.t x

{eq}\displaystyle \frac{d^2y}{dx^2}=x\frac{dy}{dx}+y-\frac{dy}{dx}\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:By\:product\:rule\:\left(uv\right)'=uv'+u'v \displaystyle \frac{d^2y}{dx^2}=\left(x-1\right)\frac{dy}{dx}+y\:\:\\ \displaystyle \frac{d^2y}{dx^2}=\left(x-1\right)\left(xy-y\right)+y\:\\ \displaystyle \frac{d^2y}{dx^2}=x^2y-2xy+2y\:\: {/eq}

There is no concave downward region.


Learn more about this topic:

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Concavity and Inflection Points on Graphs

from Math 104: Calculus

Chapter 9 / Lesson 5
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