Problem 8 - Calculus Early Transcendentals: Pearson New International Edition

Question

First find the general solution (involving a constant C) for the given differential equation. Then find the particular solution that satisfies the indicated condition. (See Example 2.) \(\frac{d y}{d x}=\sqrt{\frac{x}{y}} ; y=4\) at \(x=1\)

Step-by-Step Solution

Answer

The particular solution is \( y^{3/2} = x^{3/2} + 7 \).

Step 1: Separate Variables

The given differential equation is \( \frac{d y}{d x} = \sqrt{\frac{x}{y}} \). To solve it, we'll start by separating the variables. Rewrite it as \( y^{1/2} dy = x^{1/2} dx \).

Step 2: Integrate Both Sides

Now integrate both sides of the equation. The left side becomes \( \int y^{1/2} \, dy \), which equals \( \frac{2}{3} y^{3/2} \). The right side becomes \( \int x^{1/2} \, dx \), which equals \( \frac{2}{3} x^{3/2} \).

Step 3: Add Integration Constant

After integrating, we have \( \frac{2}{3} y^{3/2} = \frac{2}{3} x^{3/2} + C \), where \( C \) is the integration constant.

Step 4: Solve for General Solution

For simplification, multiply the entire equation by \( \frac{3}{2} \) to clear the fractions, yielding \( y^{3/2} = x^{3/2} + K \) where \( K = \frac{3}{2}C \). This is the general solution.

Step 5: Use Initial Condition to Find Particular Solution

Given the initial condition \( y = 4 \) when \( x = 1 \), substitute these values into the general solution: \( 4^{3/2} = 1^{3/2} + K \). Since \( 4^{3/2} = 8 \) and \( 1^{3/2} = 1 \), we have \( 8 = 1 + K \). So, \( K = 7 \).

Step 6: Write the Particular Solution

Substitute \( K = 7 \) back into the general solution to find the particular solution: \( y^{3/2} = x^{3/2} + 7 \).

Key Concepts

Separation of Variables

Separation of variables is a common technique used to solve differential equations. It involves rearranging the equation so that all terms containing one variable are on one side, and all terms containing the other variable are on the other side. In the example provided, the equation \( \frac{d y}{d x} = \sqrt{\frac{x}{y}} \) can be rewritten using separation of variables as \( y^{1/2} dy = x^{1/2} dx \).

This sets the stage for integrating each side separately and is crucial for finding functions that describe relationships between variables. Remember:

• Keep one variable and its derivative on one side of the equation.
• Ensure the other variable and its derivative are on the opposite side.

Integration

Integration is the process of finding the antiderivative of a function, which is essential after separating variables in a differential equation. In this context, we integrate both sides of the equation: \( \int y^{1/2} dy \) and \( \int x^{1/2} dx \).

Here's what happens next:

• The left side yields \( \frac{2}{3} y^{3/2} \).
• The right side yields \( \frac{2}{3} x^{3/2} \).

The result of the integration includes an arbitrary constant \( C \), because indefinite integrals always result in a family of functions rather than a unique solution. This constant is crucial for finding particular solutions.

Initial Value Problems

In many real-world scenarios, differential equations need to be solved under specific conditions, called initial conditions. These lead us to deal with initial value problems.

The given initial condition \( y = 4 \) when \( x = 1 \) helps us find the unique solution fitting this particular scenario. With this information, we substitute these values into the general solution:

• Substitute \( x = 1 \) and \( y = 4 \) into \( y^{3/2} = x^{3/2} + K \).
• Calculate to determine the value of \( K \).

Using initial values is critical in applications where initial circumstances are given or when a unique solution to a differential equation is required.

Particular Solution

The particular solution is a specific solution that satisfies the given initial condition. It is derived from the general solution of a differential equation.

In our situation, we derived the general solution: \( y^{3/2} = x^{3/2} + K \). Using the initial conditions, we found \( K = 7 \).

• This means the particular solution is \( y^{3/2} = x^{3/2} + 7 \).
• This specific solution fits our initial condition, making it unique to this problem.

Particular solutions are highly useful as they offer real-world applications by conforming to specific preset conditions, ensuring the models are reflective of actual scenarios.