Problem 27 - Calculus Concepts

Question

Identify the differential equation as one that can be solved using only antiderivatives or as one for which separation of variables is required. Then find a general solution for the differential equation. \(\frac{d y}{d x}=\frac{k}{x}\)

Step-by-Step Solution

Answer

The equation can be solved using antiderivatives: \( y = k \ln |x| + C \).

Step 1: Identify the Type of Differential Equation

The given differential equation is \( \frac{d y}{d x} = \frac{k}{x} \). This equation can be solved using the method of antiderivatives because the right-hand side depends only on \( x \) and not on \( y \). Hence, it is an equation that can be directly integrated.

Step 2: Integrate Both Sides

We integrate both sides of the equation with respect to \( x \). The left side becomes \( \int d y = y \) and the right side becomes \( \int \frac{k}{x} \, dx \).

Step 3: Perform the Integration

To integrate the right side, \( \int \frac{k}{x} \, dx \), we find that it equals \( k \ln |x| + C \), where \( C \) is the constant of integration.

Step 4: Write the General Solution

Combining the results from integrating both sides, the general solution to the differential equation is \( y = k \ln |x| + C \).

Step 5: Verify the Solution

Differentiate \( y = k \ln |x| + C \) with respect to \( x \) to ensure it satisfies the original differential equation. The derivative, \( \frac{d y}{d x} = \frac{k}{x} \), confirms our solution is correct.

Key Concepts

Antiderivatives

When solving differential equations, understanding antiderivatives is crucial. An antiderivative is a function that reverses the process of differentiation. To put it simply, if you have a function, the antiderivative of that function is another function whose derivative gives you the original function back.
This is particularly useful when dealing with separable equations. In the context of the original exercise, we identify the differential equation \( \frac{d y}{d x} = \frac{k}{x} \) as one that can be solved using antiderivatives alone. This is because the equation is straightforward and does not involve the dependent variable \( y \) on the right-hand side.

• The process involves recognizing that if \( \frac{d y}{d x} \) can be expressed purely in terms of one variable (like \( x \)), finding its antiderivative gives us the solution.
• Here, we found that \( y = k \ln |x| + C \) is the general solution, where \( C \) is the constant of integration. This shows the application of the antiderivative concept in finding the solution.

Understanding antiderivatives helps you solve problems by "undoing" differentiation and arriving at a function that explains the changes modeled by the equation.

Separation of Variables

Separation of variables is another core technique in solving certain types of differential equations. This method is applicable when both sides of a differential equation can be expressed as a product of functions that depend solely on one variable.
To elaborate on the concept — suppose we have an equation like \( \frac{d y}{d x} = g(x)h(y) \). Here, our task is to separate the terms involving \( y \) and \( x \) entirely onto different sides of the equality.

• In the case where it's separable, we can rewrite the equation as \( \frac{1}{h(y)} \frac{d y}{d x} = g(x) \) and subsequently integrate both sides independently.
• This approach works well when the equation's structure naturally allows terms rearrangement so each variable stands alone on one side.

However, the original exercise did not require separation of variables. This is because the equation \( \frac{d y}{d x} = \frac{k}{x} \) only involves the independent variable \( x \) on the right, making antiderivative computation direct.

Integration

Integration is the key mathematical operation used to find antiderivatives, and it plays a vital role in solving differential equations. It represents the summation of infinitesimally small changes, which accumulates to give the original quantity described by its rate of change.
In simpler terms, if differentiation is about finding how a quantity changes, integration is about finding the quantity itself. Using integration, we can solve differential equations by reversing differentiation.

• Consider the equation \( \frac{d y}{d x} = \frac{k}{x} \). We integrate both sides to find that \( \int d y = y \) and \( \int \frac{k}{x} \, dx = k \ln |x| + C \).
• These integrations give us the general solution, \( y = k \ln |x| + C \), correctly aligning with the original differential context.

While integration sometimes seems complicated, it becomes systematic through recognition of common patterns. Mastering integration methods helps significantly in addressing differential equations efficiently, whether using straightforward antiderivatives or through separation of variables.