Problem 22 - Applied Calculus

Question

Match solutions and differential equations. (Note: Each equation may have more than one solution, or no solution.) (a) \(\frac{d y}{d x}=\frac{y}{x}\) (b) \(\frac{d y}{d x}=3 \frac{y}{x}\) (c) \(\frac{d y}{d x}=3 x\) (d) \(\frac{d y}{d x}=y\) (c) \(\frac{d y}{d x}=3 y\) (I) \(y=x^{3}\) (II) \(y=3 x\) \((\text { III }) y-e^{3 x}\) (IV) \(y=3 e^{x}\) (V) \(y=x\)

Step-by-Step Solution

Answer

(a)-(V); (b)-(I); (d)-(IV); (e)-(III); (c) has no match.

Step 1: Analyze Differential Equation (a)

Equation (a) is \( \frac{dy}{dx} = \frac{y}{x} \). It is a separable equation, and solving it gives the solution in the form \( y = Cx \). This means \( y = x \) is a possible solution.

Step 2: Analyze Differential Equation (b)

Equation (b) is \( \frac{dy}{dx} = 3 \frac{y}{x} \). Solving this separable equation gives the solution in the form \( y = Cx^3 \). Hence, \( y = x^3 \) is a solution.

Step 3: Analyze Differential Equation (c)

Equation (c) is \( \frac{dy}{dx} = 3x \). Integrating with respect to \( x \) gives the solution \( y = \frac{3x^2}{2} + C \). Thus, none of the listed solutions match this result exactly.

Step 4: Analyze Differential Equation (d)

Equation (d) is \( \frac{dy}{dx} = y \). The general solution is \( y = Ce^x \). Thus, solutions with a form involving an exponential term, like \( y = 3e^x \), could fit here as they match the exponential growth characteristic.

Step 5: Analyze Differential Equation (e)

Equation (e) is \( \frac{dy}{dx} = 3y \). The general solution is \( y = Ce^{3x} \). Therefore, \( y - e^{3x} \) could match this structure depending on constants.

Step 6: Match each Differential Equation to its Solutions

- For equation (a): possible solution is (V) \( y = x \).- For equation (b): possible solution is (I) \( y = x^3 \).- For equation (c): no provided solution matches exactly.- For equation (d): possible solution is (IV) \( y = 3e^x \).- For equation (e): likely solution is (III) \( y - e^{3x} \), assuming constants are adjusted appropriately.

Key Concepts

Separable Equations

Separable equations are a type of differential equation that can be written in such a way that all terms involving one variable are on one side of the equation, and all terms involving the other variable are on the opposite side. For instance, consider the differential equation \( \frac{dy}{dx} = \frac{y}{x} \). It can be rearranged to:

• \( x \, dy = y \, dx \)

Here, each variable is isolated on different sides of the equation, making it possible to integrate both sides separately. This is the typical process of solving separable equations. It is a method that simplifies finding a particular or general solution to differential equations, often involving integration once both sides are separated.

Exponential Growth

Exponential growth describes a process where the rate of change of a quantity is proportional to the current quantity itself. This often appears in differential equations where rates of change are expressed as directly proportional to the current amount or population. The equation \( \frac{dy}{dx} = y \) is a classic example representing exponential growth. When solved, the general solution is \( y = Ce^x \), where \( C \) is a constant. This constant is determined by specific initial conditions. In these scenarios, the solution demonstrates how the quantity grows exponentially over time, showing the power of exponential functions in modeling real-world phenomena such as population dynamics and radioactive decay.

Integration

Integration is a fundamental concept in calculus often used to solve differential equations. The process of integration involves finding the antiderivative of a function, which essentially reverses differentiation. Consider the differential equation \( \frac{dy}{dx} = 3x \). Solving it requires integrating both sides:

• \( y = \int 3x \, dx \)
• Resulting in \( y = \frac{3x^2}{2} + C \)

Here, \( C \) represents an arbitrary constant that arises from the indefinite integration. Integration is crucial in obtaining solutions to differential equations, allowing for analysis of the system behavior described by the differential equations over a continuous range rather than at a discrete set of points.

General Solution

In the context of differential equations, the general solution refers to a solution that contains all possible solutions of the differential equation, often expressed in terms of an arbitrary constant. For example, when dealing with the differential equation \( \frac{dy}{dx} = 3y \), its general solution is given by \( y = Ce^{3x} \). The arbitrary constant \( C \) allows the solution to fit multiple initial conditions or specific scenarios. The beauty of the general solution lies in its ability to encompass a multitude of specific solutions, serving as a comprehensive form from which particular solutions can be derived by assigning specific values to \( C \) based on boundary conditions or initial values.