Problem 50 - Calculus and its applications

Question

(a) write a differential equation that models the situation, and (b) find the general solution. If an initial condition is given, find the particular solution. Recall that when \(y\) is directly proportional to \(x,\) we have \(y=k x\), and when \(y\) is inversely proportional to \(x,\) we have \(y=k / x,\) where \(k\) is the constant of proportionality. In these exercises, let \(k=1\). The rate of change of \(y\) with respect to \(x\) is inversely proportional to the square root of \(y\).

Step-by-Step Solution

Answer

The differential equation is \(\frac{dy}{dx} = \frac{1}{\sqrt{y}}\), and the general solution is \(\frac{2}{3}y^{3/2} = x + C\).

Step 1: Understanding the Problem

We need to write a differential equation representing the given situation. The problem states that the rate of change of \(y\) with respect to \(x\) is inversely proportional to the square root of \(y\). This means \(\frac{dy}{dx} = \frac{k}{\sqrt{y}}\) where \(k\) is a constant.

Step 2: Setting the Proportionality Constant

From the problem, we know that the constant of proportionality \(k = 1\). Therefore, we substitute \(k\) into the equation: \[\frac{dy}{dx} = \frac{1}{\sqrt{y}}.\] This is the differential equation that models the given situation.

Step 3: Solving the Differential Equation

To solve the differential equation, we separate the variables. Rearrange the equation to isolate terms involving \(y\) on one side and terms involving \(x\) on the other: \[\sqrt{y} \cdot dy = dx.\] Now, integrate both sides: \[\int \sqrt{y} \, dy = \int \, dx.\]

Step 4: Performing the Integration

Integrate both sides: \[\int \sqrt{y} \, dy = \frac{2}{3}y^{3/2} + C_1\] and \[\int \, dx = x + C_2.\] Equating both expressions gives us: \[\frac{2}{3}y^{3/2} + C_1 = x + C_2.\]

Step 5: Finding the General Solution

To simplify, combine constants into a single constant: \[\frac{2}{3}y^{3/2} = x + C.\] where \(C = C_2 - C_1\). This equation represents the general solution to the differential equation.

Step 6: Finding the Particular Solution (if Initial Condition is Given)

If an initial condition \(y(x_0) = y_0\) is provided, substitute \(x_0\) and \(y_0\) into \(\frac{2}{3}y^{3/2} = x + C\) to find \(C\). Then substitute back to get the particular solution. Since no initial condition is provided, we stop here.

Key Concepts

Proportional Relationships

In the realm of differential equations, proportional relationships form the backbone of many situations that involve changes and dependencies between two quantities.
In simpler terms, if we say that something is directly proportional to another quantity, it means that as one quantity increases, the other one increases at a constant rate, multiplied by a fixed number we call the "constant of proportionality." In our scenario, when we mention "the rate of change of y with respect to x is inversely proportional to the square root of y," it pinpoints an inverse proportionality relationship.
This implies that as one quantity (in this case, \(\frac{dy}{dx}\)) increases, the other quantity, \(\sqrt{y}\), decreases, and vice versa. The constant of proportionality bridges these two quantities.
The magical part of such equations is how proportionality and constants have the power to precisely model complex real-world phenomena with elegantly simple equations. In exercises like these, understanding the setup of proportional relationships aids us in crafting solutions that predict behaviors and outcomes.

General Solution

When tackling a differential equation, the general solution is kind of like the broad blueprint or the map that shows all possible solutions for the equation.
Let's break it down: our differential equation is \(\frac{dy}{dx} = \frac{1}{\sqrt{y}}\), which describes how the change in \(y\) is connected to \(x\). This equation is our starting point. We solve this to uncover the general path \(y\) will follow.To find the general solution, we carry out something called separation of variables, which means we rearrange the equation so each side contains only one variable (either \(x\) or \(y\)). Then, we perform integration on both sides. This means finding an antiderivative - kind of like reverse differentiation.
For our specific case, the integration leads to the formula \[\frac{2}{3}y^{3/2} = x + C\]where \(C\) stands for a constant.
This constant represents an infinite number of values, making the solution 'general' because it can fit an infinite variety of situations governed by the same differential equation. The beauty of general solutions lies in their ability to adapt to many scenarios, with each different value of \(C\) corresponding to a unique situation.

Particular Solution

While the general solution offers a broad range of possibilities, the particular solution pinpoints one specific path out of all the possible solutions.
To find a particular solution, we need more information, namely an initial condition. This is usually given in the form of a specific point through which the solution passes, represented as \(y(x_0) = y_0\). In practice, acquiring a particular solution means taking the general formula and solving for the constant \(C\) using the initial point. After finding this unique constant value, we implement it back into our general equation, filling in the blanks with the provided data points.
If, for instance, we had been given that when \(x = 1\), \(y = 2\), we would substitute these values into our general solution to calculate \(C\), and hence, derive the particular formula.
It's like saying, "Out of all potential roads mapped out by our general equation, this is the single road traveled, considering these specific starting conditions." However, in our case, without designated initial conditions, we conclude with the general solution because we lack sufficient information to pinpoint a particular path.