Problem 28 - Advanced Engineering Mathematics

Question

$$ 3 y^{\prime}=4 y $$

Step-by-Step Solution

Answer

The solution is \(y = Ce^{\frac{4}{3}x}\), where \(C\) is a constant.

Step 1: Understand the Problem

The given problem is a first-order linear differential equation: \(3y' = 4y\). Here, \(y'\) denotes the derivative of \(y\) with respect to \(x\). Our objective is to find the function \(y(x)\).

Step 2: Rearrange the Equation

Start by rearranging the equation to isolate terms involving \(y'\) and \(y\). Divide both sides by 3: \(y' = \frac{4}{3}y\).

Step 3: Separate Variables

Separate the variables to facilitate integration. Rearrange to get: \(\frac{dy}{y} = \frac{4}{3}dx\).

Step 4: Integrate Both Sides

Integrate both sides of the separated equation. Integrate the left side with respect to \(y\), and the right side with respect to \(x\): \(\int \frac{1}{y} \, dy = \int \frac{4}{3} \, dx\). This gives: \(\ln |y| = \frac{4}{3}x + C\), where \(C\) is the integration constant.

Step 5: Solve for \(y\)

To solve for \(y\), exponentiate both sides to remove the natural logarithm: \(e^{\ln |y|} = e^{\frac{4}{3}x + C}\). This simplifies to: \(|y| = e^{C}e^{\frac{4}{3}x}\). Let \(e^{C} = C_1\) (a positive constant), thus: \(y = C_1e^{\frac{4}{3}x}\). Since \(y\) could also be negative, write the solution as: \(y = Ce^{\frac{4}{3}x}\), where \(C\) can be any real constant.

Key Concepts

Differential Equation Solution Steps

Solving differential equations involves a series of logical steps that allow us to find the unknown function. When dealing with first-order linear differential equations, like the one given in the problem, the aim is to determine the function \( y(x) \). The process is systematic to ensure an accurate solution.

Here's a concise breakdown of the procedure:

• Identify and understand the type of differential equation.
• Rearrange the equation in standard form \( y' + p(x)y = q(x) \) if necessary.
• Apply the appropriate method, such as separation of variables, to solve the equation.
• Integrate to find the general solution.
• Finally, consider any initial conditions or additional information to find a particular solution.

For our given equation \( 3y' = 4y \), we followed these steps, eventually reaching the general solution \( y = Ce^{\frac{4}{3}x} \). Each step has its importance, ensuring that all variables are properly handled and integrated.

Variable Separation Method

The variable separation method is a fundamental technique to tackle differential equations. It allows splitting the variables, making them easier to integrate.

In essence, the goal is to rearrange the equation so that each integration can be performed independently with respect to a single variable. Here's how it works step-by-step:

• Start by rearranging the differential equation to separate \( y \) and \( x \) so that all terms involving \( y \) are on one side, and all terms involving \( x \) are on the other. For our example, we rewrite the equation as \( \frac{dy}{y} = \frac{4}{3} dx \).
• Now, both sides are ready for integration. This allows us to integrate the left side just with respect to \( y \), and the right side solely in terms of \( x \).
• This separation makes the equation solvable, leading to a solution that reflects the integration of each side separately.

By isolating the variables, we use simpler integration techniques to reach the solution. This method is particularly useful for first-order differential equations like the one we solved.

Integration of Differential Equations

Once the variables are separated in a differential equation, integration is the next crucial step. This process is where the separated parts of the equation each get integrated with respect to their respective variables.

For the equation \( \frac{dy}{y} = \frac{4}{3} dx \), integration is performed as follows:

• The left-hand side becomes \( \int \frac{1}{y} \, dy \), which integrates to \( \ln |y| \).
• The right-hand side is \( \int \frac{4}{3} \, dx \), simplifying to \( \frac{4}{3}x + C \), where \( C \) is the integration constant.
• Combining these results yields: \( \ln |y| = \frac{4}{3}x + C\).
• Solving for \( y \) involves exponentiating both sides, yielding \( y = Ce^{\frac{4}{3}x} \).

Integration is key as it transforms the separated variables into a cohesive solution. The constant \( C \) captures any shifts or transformations, mirroring real-world conditions or additional data inferred from situations involving the differential equation.