Problem 33 - Calculus Concepts

Question

For Activities 33 through \(36,\) write a differential equation expressing the information given and, when possible, find a general solution for the differential equation. The Rowan-Robinson model of the universe assumes that the universe is expanding with respect to time \(t\) at a rate that is decreasing in inverse proportion to the square of its current size \(S .\)

Step-by-Step Solution

Answer

The differential equation is \(\frac{dS}{dt} = \frac{k}{S^2}\). The general solution is \(S = \sqrt[3]{3kt + 3C}\).

Step 1: Understand the problem

The problem states that the universe is expanding at a rate that decreases inversely with the square of its size. We need to express this relationship in a differential equation.

Step 2: Formulate the relationship

The expansion rate of the universe, i.e., the rate of change of size \(S\) with respect to time \(t\), is given by \(\frac{dS}{dt}\). According to the problem, this rate is inversely proportional to \(S^2\). Mathematically, this is expressed as \(\frac{dS}{dt} = \frac{k}{S^2}\), where \(k\) is a constant of proportionality.

Step 3: Solve the differential equation

Reorganize the differential equation to separate the variables. This gives \(S^2 \, dS = k \, dt\). Integrate both sides: \(\int S^2 \, dS = \int k \, dt\). The integrals yield \(\frac{S^3}{3} = kt + C\), where \(C\) is an integration constant.

Step 4: Find the general solution

Solve for \(S\) to find the general solution. Multiply both sides by 3 to get \(S^3 = 3kt + 3C\). Then, solve for \(S\): \(S = \sqrt[3]{3kt + 3C}\). This is the general solution expressing the size \(S\) of the universe concerning time \(t\).

Key Concepts

Universe Expansion Model

The universe expansion model refers to the way cosmologists explain how the universe increases its size over time. A prominent theory behind this idea is that the universe's expansion rate changes as a function of its size and time. In our exercise, the expansion rate is not constant but decreases over time, following an inverse relationship with the square of its current size, \(S\). This means that as the universe expands and \(S\) becomes larger, the rate of its expansion \(\frac{dS}{dt}\) slows down. This model provides crucial insights into cosmological studies and helps describe the dynamic nature of the universe.

Proportional Relationships

Proportional relationships describe how one quantity changes in relation to another. In the context of our problem, the rate of expansion \(\frac{dS}{dt}\) is inversely proportional to the square of the universe's size, \(S^2\). Mathematically, this is represented by the differential equation \(\frac{dS}{dt} = \frac{k}{S^2}\), where \(k\) is a constant. This indicates that as the universe grows larger, the rate of expansion lessens, since the rate is divided by a larger number. It's a fundamental concept in mathematics and physics, allowing us to model and understand complex natural phenomena through simple relationships.

Integration Constants

When solving differential equations, integration constants are parameters that arise during the integration process. When we integrate the equation \(\int S^2 \, dS = \int k \, dt\), we obtain \(\frac{S^3}{3} = kt + C\), where \(C\) is the integration constant. This constant represents an unknown that must be determined by initial conditions or boundary conditions. Every time we perform an integration operation, an integration constant emerges because integration, essentially an anti-differentiation process, is not unique without additional information to pin down the constant value.

Separable Differential Equations

Separable differential equations are a type of differential equation that can be rewritten so that each variable and its differentials appear on opposite sides of the equation. In our example, the initial equation \(\frac{dS}{dt} = \frac{k}{S^2}\) is transformed to separate \(S\) and \(t\) into \(S^2 \, dS = k \, dt\). This makes it easier to integrate both sides independently. Separable differential equations are one of the simplest forms of differential equations to solve and are ubiquitously used to model real-world phenomena, from physics to biology, helping to simplify complex systems into manageable mathematical problems.