Problem 14 - Thomas Calculus

Question

Solve the differential equation in Exercises \(9-18\). $$ \sqrt{2 x y} \frac{d y}{d x}=1 $$

Step-by-Step Solution

Answer

The solution is \( \frac{2}{3} (2xy)^{3/2} = x + C \).

Step 1: Rearrange the Equation

First, we start with the given differential equation: \( \sqrt{2xy} \frac{dy}{dx} = 1 \). We can simplify this by separating variables. Rearrange it to \( \frac{dy}{dx} = \frac{1}{\sqrt{2xy}} \).

Step 2: Separate the Variables

Next, we separate the variables. The equation becomes \( \sqrt{2xy} \, dy = dx \). The \( y \) terms are on one side and the \( x \) terms on the other side.

Step 3: Integrate Both Sides

Now, integrate both sides of the equation. The left side becomes \( \int \sqrt{2xy} \, dy \), while the right side becomes \( \int dx \). Before integrating the left side, recognize that it requires substitution or is evaluated using a power rule.

Step 4: Substitution in Integration

For the left integral, set \( u = 2xy \), then \( du = 2x \, dy \), so \( dy = \frac{du}{2x} \). Substitute \( u \) into the integral, changing it to \( \int \sqrt{u} \frac{du}{2x} = \int \frac{1}{2x} u^{1/2} \, du \).

Step 5: Solve the Integrals

The integral \( \int \frac{1}{2x} u^{1/2} \, du \) depends on \(x\) being constant relative to \(du\). Solving both, we get \( \frac{2}{3(2x)} u^{3/2} \) for the left side and \( x + C \) for the right side, where \( C \) is the constant of integration.

Step 6: Substitute Back and Solve

Substitute back \( u = 2xy \) into the solution, obtaining \( \frac{2}{3} (2xy)^{3/2} = x + C \). This represents a general solution to the differential equation.

Step 7: Simplify the Solution

Simplify the equation: \( \frac{2}{3} (2xy)^{3/2} = x + C \). The solution may generally be left in this implicit form or solved explicitly for \( y \), if desired.

Key Concepts

Separation of Variables

Separation of variables is a technique used to simplify solving differential equations by rearranging them so each variable and its derivative are on opposite sides of the equation. This method breaks the problem into smaller parts that are often easier to solve.
In the given exercise, the equation \(\sqrt{2xy} \frac{dy}{dx} = 1\) was transformed. By dividing both sides by \(\sqrt{2xy}\) and multiplying by \(dx\), we isolate the \(y\) terms on one side and the \(x\) terms on the other.
This resulted in the equation \(\sqrt{2xy} \, dy = dx\). Now, we can focus on each variable separately during the integration process.

Integration Techniques

Integration techniques are crucial for solving differential equations, especially when integrating more complex functions. Once we have separated the variables, each side of the equation needs to be integrated with respect to its variable.
In our exercise, we encountered the integral \(\int \sqrt{2xy} \, dy\). To solve this, a substitution method was employed.

• Substituted \(u = 2xy\), which led to \(du = 2x \, dy\). Thus, \(dy = \frac{du}{2x}\).
• This substitution simplifies the integral into a power formula: \(\int \frac{1}{2x} u^{1/2} \, du\).

These techniques help manage difficult integrals by converting them into forms that are easier to integrate.

General Solution

The general solution represents the family of all possible solutions to a differential equation. It includes an arbitrary constant, indicating there are infinitely many solutions.
After carrying out integration for both sides of the equation, we arrived at \(\frac{2}{3} (2xy)^{3/2} = x + C\). This is a form of the general solution for our differential equation.
This equation describes a relationship between \(x\) and \(y\) that satisfies the original differential equation for any value of the constant \(C\). Such solutions may be left in implicit forms or solved to express \(y\) explicitly if necessary.

Constant of Integration

The constant of integration, symbolized as \(C\), appears in the solution of a differential equation due to indefinite integration. This constant represents any vertical shift in the graph of an antiderivative, introducing an arbitrary solution component.
When we integrated both sides of the separated equation, \(C\) appeared in the final expression \(\frac{2}{3} (2xy)^{3/2} = x + C\).
This constant is essential, as it allows the general solution to represent a whole family of solutions. Without specific boundary or initial conditions, \(C\) remains undetermined, illustrating the infinite variety of functions that possess identical derivatives.