The registrar of Kellogg University estimates that the...

The registrar of Kellogg University estimates that the student enrollment in the Continuing Education division will grow at the rate of $$N^{\prime}(t)=2000(1+0.2 t)^{-3 / 2}$$ students/year, $t$ years from now. If the current student enrollment is 1000 , find an expression giving the enrollment $t$ years from now. What will be the enrollment 5 years from now?

Key Concepts

Substitution Method (u-substitution)

The substitution method, or u-substitution, is a technique used in integration to simplify an integral by changing variables. By substituting part of the integrand with a new variable, the integral often becomes easier to evaluate. This technique is particularly useful when dealing with composite functions or when the integrand's structure suggests a convenient change of variables.

Integration

Integration is the mathematical process used to find the original function given its derivative. In the context of a differential equation, integration is necessary to recover the function that models the quantity of interest. Evaluating the integral while applying the correct limits or constants yields the function that describes the system over time.

Growth Rate

The growth rate describes how a quantity changes over time and is represented mathematically by a derivative. In many real-world models, including enrollment or population growth, understanding the rate at which the quantity increases or decreases is critical for predicting future behavior and making informed decisions.

Differential Equations

Differential equations involve equations that relate a function to its derivatives. They are used to describe continuous change and are fundamental in modeling various phenomena, including growth rates, decay processes, and dynamics in physics, biology, economics, and engineering. The problem involves a differential equation that expresses the rate of change of enrollment over time.

Initial Value Problems

An initial value problem specifies not only a differential equation but also an initial condition, which uniquely determines a solution. The initial value is used to integrate the differential equation correctly, ensuring that the model accurately reflects the starting state of the system, such as a given enrollment level at the current time.

Transcript

00:01 Hello, welcome to this lesson.

00:02 So this equation represents the rate of attendance, the rate of enrollment.

00:11 Student, with respect to timing, yes.

00:15 We are looking for the number of enrollment.

00:19 So in this sense, we would integrate this so that we can get the number of enrollment.

00:27 Is to have in the rate so this is equal to the integral both sides respect to t.

00:48 Now we'll make a substitution one plus zero point so that you have the u that is equal to 0 .2.

01:01 Then i would have the t cost to b u all over 0 .2 2 so you'll substitute this and you substitute that.

01:15 Now we have the number of enrollment that is equal.

01:24 Now in u.

01:26 So you have 2 ,000 plus 0 .2t now becomes a u.

01:36 You have 3 to the power 3 on 2.

01:42 So dt becomes 0 .2.

01:46 So instead of it you put in the u all over zero point so here we have enough you again that is equal to come bring 10 ,000 outside that is 2 ,000 divided by 0 .2 we have this so if we integrate 10 ,000 then we have u to the power half all over 1, all over negative half, then plus c.

02:41 So this becomes 20 ,000, negative 20 ,000 u to the power half plus c.

02:53 So you can write that's negative 20 ,000 all over square roots of you, then plus c.

03:02 Now we can write everything in terms of t...

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