# Video: Identifying a Differential Equation That Represents Linear Decay

If π(π‘) is the number of unemployed people at time π‘ in a country, which of the following differential equations describes linear decay in the number of unemployed people? [A] dπ/dπ‘ = β150π [B] dπ/dπ‘ = β300π‘ [C] dπ/dπ‘ = β300 πΒ² [D] dπ/dπ‘ = β150 π‘Β² [E] dπ/dπ‘ = β300

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### Video Transcript

If π of π‘ is the number of unemployed people at time π‘ in a country, which of the following differential equations describes linear decay in the number of unemployed people? (a) dπ by dπ‘ is negative 150 times π. (b) dπ by dπ‘ is negative 300 times π‘. (c) dπ by dπ‘ is negative 300 times π squared. (d) dπ by dπ‘ is negative 150 times π‘ squared. Or (e) dπ by dπ‘ is negative 300.

To determine which of the differential equation describes linear decay in the number of unemployed people, we need to pull relevant information from the question. Weβre told that the number of unemployed people at time π‘ in a country is described by the function π of π‘. And this means that the number of unemployed people changes with time. Weβre also told that the change in the number of unemployed people takes the form of linear decay.

We know that if a function is linear, it follows a straight line and that, in mathematical terms, decay refers to decreasing. If we sketch this behavior then, our function is a straight line with a negative slope. And as π‘ increases, the number of unemployed people falls or decreases. And symbolically, our function has the form of the equation of a straight line. That is, π is equal to π times π‘, which is our variable, plus π.

π is the slope of our line. And π is the π¦-intercept. We know that our slope π is negative since weβre talking about linear decay. And remember, weβre trying to find which differential equation describes this linear decay. All of our differential equations have the first derivative on the left-hand side. So these are first-order differential equations. And if we differentiate our function π with respect to time π‘, we have dπ by dπ‘ equal to π, which we know is a negative constant.

Remember that linear decay describes a constant decrease in values. And thatβs our slope π. Since π is negative, letβs call it a negative π, where π is a real number greater than zero. So dπ by dπ‘ is negative π. So now, letβs compare this with our differential equations. In equation (a), dπ by dπ‘ is negative 150 times π. Comparing this to our differential equation, there is a negative constant, negative 150. However, this is multiplying π in equation (a). Our equation has no π on the right-hand side. So we can discount equation (a) since this doesnβt match our differential equation.

Now, letβs look at equation (b). This has dπ by dπ‘ equal to negative 300 times π‘. Again, we have a negative constant, negative 300. And this time, our constant multiplies π‘. And again, this doesnβt match our differential equation because our differential equation has no π‘ on the right-hand side. So we can discount equation (b).

And in fact, we have a similar situation for equation (c) and equation (d). Equation (c) has a negative constant times π squared on the right-hand side. And equation (d) has a negative constant times π‘ squared on the right-hand side. Our differential equation has neither π squared nor π‘ squared on the right-hand side. So we can discount both equation (c) and equation (d).

This leaves us with equation (e), where dπ by dπ‘ is equal to negative 300. And if we let π equal to 300, then this does match with our equation. On the left-hand side, we have dπ by dπ‘. And on the right-hand side, we have a negative constant. The differential equation describing linear decay in the number of unemployed people is therefore equation (e) dπ by dπ‘ is negative 300.