Problem 24 - Calculus Concepts

Question

For Activities 24 through \(30,\) 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. \(\frac{d y}{d x}=k x\)

Step-by-Step Solution

Answer

The differential equation can be solved using antiderivatives, yielding solution: \( y = \frac{k}{2}x^2 + C \).

Step 1: Identify the Type of Differential Equation

The given differential equation is \( \frac{d y}{d x} = kx \), where \( k \) is a constant. This equation can be solved by direct integration (finding antiderivatives), as it is already expressed as \( \frac{d y}{d x} = f(x) \).

Step 2: Integrate Both Sides

To solve \( \frac{d y}{d x} = kx \), integrate both sides with respect to \( x \). The left side becomes \( y \) after integration, and the right side becomes \( \int kx \, dx \).

Step 3: Compute the Antiderivative

The antiderivative of \( kx \) is \( \frac{k}{2}x^2 + C \), where \( C \) is the constant of integration. Thus, the equation becomes \( y = \frac{k}{2}x^2 + C \).

Step 4: Write the General Solution

The general solution of the differential equation \( \frac{d y}{d x} = kx \) is \( y = \frac{k}{2}x^2 + C \), where \( C \) is any real number.

Key Concepts

Antiderivatives

When dealing with differential equations, finding antiderivatives is a method commonly used to solve equations like \( \frac{d y}{d x} = kx \). The term 'antiderivative' refers to a function that reverses the process of differentiation. In simpler terms, if differentiation finds the slope of a function, an antiderivative will help locate the original function based on this slope.
To identify whether a given differential equation can be solved with antiderivatives, check if it can be expressed as \( \frac{d y}{d x} = f(x) \). If it is in this form, solving it simply requires integrating the function on the right side to find the original function.

• Integrate the function: Solving \( \frac{d y}{d x} = kx \) involves integrating \( kx \), which results in the antiderivative.
• Constant of Integration: Remember to add a constant \( C \), representing the fact that there are infinitely many antiderivatives, because adding any constant will not change the derivative.

Practicing with antiderivatives is crucial because they are a fundamental tool used frequently in calculus and solving differential equations.

Separation of Variables

The method of separation of variables comes into play when a differential equation isn’t straightforward to integrate directly. However, in this specific exercise, the separation of variables method isn’t required. It is beneficial, though, to understand when this method is applicable.Separation of variables is typically used when a differential equation can be rewritten to isolate all the \( y \) terms on one side and all the \( x \) terms on the other. This layout allows us to integrate each side with respect to its variable, aiding in the discovery of the solution.

• Rewriting Equations: The goal is to manipulate the equation into the form \( g(y)dy = f(x)dx \), where the left side can be integrated in terms of \( y \) and the right side in terms of \( x \).
• Applicable Scenarios: This method is not employed for the equation in this exercise because \( \frac{d y}{d x} = kx \) is already simple enough to be solved with antiderivatives.

Understanding separation of variables is important for more complex differential equations where other simpler methods don't apply.

General Solution

The general solution of a differential equation represents a family of solutions that encompasses all particular answers for the situation defined by the equation. For the given problem \( \frac{d y}{d x} = kx \), the solution found was a general solution.The general solution is expressed as \( y = \frac{k}{2}x^2 + C \). Here, \( C \) is the constant of integration that can take any real value. This constant is vital because:

• Flexibility: It indicates the array of possible functions (the entire family) that can satisfy the differential equation.
• Initial Conditions: If additional specific information is provided, such as a point \( (x_0, y_0) \) through which the solution must pass, \( C \) can be determined to find a particular solution.

In real-world scenarios, having the general solution allows for flexibility in modeling diverse situations that the mathematical model may describe. Recognizing the role of \( C \) in the general solution is essential for effectively applying mathematics to solve practical problems.