Problem 4 - Calculus: An Integrated Approach to Functions and Their Rates of Change

Question

Solve the given differential equation. \(\frac{d y}{d x}=\frac{y}{x}\)

Step-by-Step Solution

Answer

The solution to the given differential equation is \(y = (x^2/2 - c) / x\)

Step 1: Separate the Variables

Separate the variables by multiplying both sides of the equation by \(dx\) and dividing both sides of the equation by \(y\), which gives: \(x dy = y dx\)

Step 2: Integrate both Sides

Integrate both sides with respect to their respective variables. This gives: \(\int x\, dy = \int y\, dx\)

Step 3: Solve the Integrals

Solve the integrals on both sides to get: \(x^2/2 = yx + c\) where c is the constant of integration.

Step 4: Final Solution

Solve the equation for y to get the general solution to the differential equation. The general solution is: \(y = (x^2/2 - c) / x\)

Key Concepts

Separation of Variables

In dealing with differential equations, the separation of variables is a powerful technique. This method is particularly useful for solving first-order differential equations. The main idea is to rewrite the equation in such a way that each side of the equation contains a single variable, along with its differential.

For example, consider the differential equation \(\frac{dy}{dx} = \frac{y}{x}\). To employ separation of variables, we wish to isolate \(y\) on one side and \(x\) on the other. We multiply both sides by \(dx\) and divide by \(y\). This gives us \(x dy = y dx\).

This separation sets the stage for the next step where we can integrate both sides independently. The beauty of this method is that it transforms a single differential equation into two separate integrals. This makes it manageable to solve as each variable is dealt with in its own context.

Integration

Once variables are properly separated in the differential equation, the next step is to integrate both sides. Integration is essentially finding the antiderivative of a function. It helps us compare the accumulated quantities described by the differential equations.

In our ongoing example \(x dy = y dx\), integration is applied to both sides:

• Integrate the left side with respect to \(x\): \(\int x\, dy\)
• Integrate the right side with respect to \(y\): \(\int y\, dx\)

This results in \(\frac{x^2}{2}\) for the \(x\) side, and \(yx\) for the \(y\) side.

After integrating, we always include a constant of integration, \(c\), since the integral of a function can vary by a constant. It is crucial to remember these constants to ensure the completeness of the solution.

General Solution

The general solution refers to the form of the solution of a differential equation that encompasses all possible particular solutions. It usually involves an arbitrary constant, reflecting the infinitely many solutions that differ by this constant.

In the example equation, after integrating and simplifying as explained earlier, we arrive at equation \(\frac{x^2}{2} = yx + c\). Solving this for \(y\), we find:

\[y = \frac{x^2/2 - c}{x}\]

This expression represents the general solution. By adjusting the constant \(c\), we can generate a family of functions that will solve the original differential equation for different initial conditions. This flexibility is what makes differential equations such a versatile tool in mathematical modeling. Recognizing how the constant \(c\) influences the solution is essential for understanding the full breadth of solutions available for a given problem.