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Write a differential equation for the following: a cell starts at a volume of 400m^3 and gains...

Question:

Write a differential equation for the following: a cell starts at a volume of 400m{eq}^3 {/eq} and gains volume at a rate of 3m{eq}^3 {/eq}/s.

Differential Equations:

Differential equations are equations that contain derivatives as their terms. Differential equations can be formed by differentiating the function and arranging it in a suitable manner. Often, we will use the power rule of differentiation, which is given by {eq}\displaystyle \frac{d}{dx} x^n=nx^{n-1} {/eq}.

Answer and Explanation: 1

Let {eq}C(t) {/eq} be the volume of the cell present at any time {eq}t {/eq} (where the units of time are in seconds).

By the given conditions, we have an initial volume of 400m{eq}^3 {/eq} and also know that the cell gains volume at a rate of 3m{eq}^3 {/eq}/s.

{eq}\begin{align*} C(t)&=3t+400\\ \frac{dC}{dt}&=3&\text{[Differentiating with respect to t] } \end{align*} {/eq}

Note that {eq}\frac{dC}{dt} =3 {/eq} doesn't tell us much on its own,

so when giving the differential equation we typically include the initial condition as follows:

{eq}\boxed{ \frac{dC}{dt} =3, \qquad C(0) = 400 } {/eq}


Learn more about this topic:

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First-Order Linear Differential Equations

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Chapter 16 / Lesson 3
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In this lesson you'll learn how to solve a first-order linear differential equation. We first define what such an equation is, and then we give the algorithm for solving one of that form. Specific examples follow the more general description of the method.


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