Solving The Differential Equation Dy/dx + 3y = 6 A Step-by-Step Guide

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Hey everyone! Today, we're diving into the fascinating world of differential equations, specifically focusing on a first-order differential equation. We'll tackle the equation dy/dx + 3y = 6, with the initial condition y(0) = 2. Our goal is to find the general solution and understand the steps involved. So, buckle up, and let's get started!

Understanding First-Order Differential Equations

Before we jump into solving our specific equation, let's take a moment to understand what a first-order differential equation is. In simple terms, a first-order differential equation is an equation that involves a function and its first derivative. It's called "first-order" because the highest derivative present in the equation is the first derivative. These equations pop up in various fields like physics, engineering, economics, and biology, modeling phenomena that change over time or space. Think about the decay of a radioactive substance, the growth of a population, or the motion of an object – these can often be described using first-order differential equations.

Why are these equations so important? Well, they help us understand how things change! Imagine you're trying to predict the temperature of a cup of coffee as it sits on your desk. The rate at which the coffee cools down depends on the current temperature of the coffee and the temperature of the room. This relationship can be expressed as a first-order differential equation. By solving this equation, we can find a function that tells us the temperature of the coffee at any given time. This predictive power is what makes differential equations so valuable.

Types of First-Order Differential Equations: First-order differential equations come in different flavors, each requiring a slightly different approach to solve. Some common types include:

  • Separable equations: These are equations where we can separate the variables (y and x) on different sides of the equation. This makes them relatively straightforward to solve by integrating both sides.
  • Linear equations: These equations have a specific form (like the one we're tackling today) and can be solved using an integrating factor.
  • Exact equations: These equations satisfy a certain condition that allows us to find a potential function and solve the equation.
  • Homogeneous equations: These equations have a special property where the equation remains unchanged if we scale the variables by a constant factor.

Our equation, dy/dx + 3y = 6, falls into the category of linear first-order differential equations. These equations have the general form dy/dx + P(x)y = Q(x), where P(x) and Q(x) are functions of x. In our case, P(x) = 3 and Q(x) = 6. Now that we know what kind of equation we're dealing with, let's move on to the solution!

Solving the Differential Equation dy/dx + 3y = 6

Okay, let's get down to business and solve this differential equation! As we discussed, this is a linear first-order differential equation, so we'll use the method of integrating factors. This method is a powerful technique that transforms the equation into a form that we can easily integrate. The general approach involves a few key steps:

  1. Find the integrating factor: The integrating factor, often denoted by μ(x), is a function that we multiply both sides of the equation by. This factor is carefully chosen so that the left-hand side of the equation becomes the derivative of a product. The formula for the integrating factor is μ(x) = e^(∫P(x) dx), where P(x) is the coefficient of y in our differential equation. In our case, P(x) = 3, so the integrating factor is:

    μ(x) = e^(∫3 dx) = e^(3x)

    So, our integrating factor is e^(3x). This is a crucial step, so make sure you understand how we got here!

  2. Multiply both sides of the equation by the integrating factor: Now, we multiply both sides of our original differential equation (dy/dx + 3y = 6) by the integrating factor e^(3x):

    e^(3x) (dy/dx + 3y) = 6e^(3x)

    This step is where the magic happens! Notice how the left-hand side is now starting to look like the result of a product rule derivative.

  3. Recognize the left-hand side as the derivative of a product: The left-hand side of the equation can be rewritten as the derivative of the product of y and the integrating factor:

    d/dx (y * e^(3x)) = e^(3x) (dy/dx) + 3e^(3x)y

    This is the key insight of the integrating factor method. We've transformed the left-hand side into a simple derivative, which we can easily integrate.

  4. Integrate both sides with respect to x: Now, we integrate both sides of the equation with respect to x:

    ∫ d/dx (y * e^(3x)) dx = ∫ 6e^(3x) dx

    The integral on the left-hand side simply undoes the derivative, leaving us with:

    y * e^(3x) = ∫ 6e^(3x) dx

    To evaluate the integral on the right-hand side, we use a simple substitution (or recognize it as a standard integral):

    ∫ 6e^(3x) dx = 2e^(3x) + C

    where C is the constant of integration. Don't forget this constant! It's essential for finding the general solution.

  5. Solve for y: Finally, we solve for y by dividing both sides of the equation by the integrating factor e^(3x):

    y = (2e^(3x) + C) / e^(3x)

    y = 2 + Ce^(-3x)

    This is the general solution to the differential equation. It represents a family of solutions, each corresponding to a different value of the constant C.

Applying the Initial Condition y(0) = 2

We've found the general solution, but we have an initial condition: y(0) = 2. This means that when x = 0, y = 2. We can use this information to find the specific value of the constant C and obtain the particular solution that satisfies this condition. Let's plug in x = 0 and y = 2 into our general solution:

2 = 2 + Ce^(-3 * 0)

2 = 2 + C * e^(0)

2 = 2 + C * 1

2 = 2 + C

From this, we can see that C = 0. So, the particular solution that satisfies the initial condition is:

y = 2 + 0 * e^(-3x)

y = 2

Wait a minute! This seems a bit simpler than we expected. The particular solution is simply y = 2. This means that the function that satisfies the differential equation and the initial condition is a constant function.

Analyzing the Solution

So, we've found that the solution to the differential equation dy/dx + 3y = 6 with the initial condition y(0) = 2 is y = 2. Let's take a moment to understand what this means. The solution y = 2 is a constant function. This means that the value of y remains constant at 2, regardless of the value of x. Geometrically, this represents a horizontal line at y = 2 on a graph.

Why is the solution a constant function? Let's go back to the original differential equation: dy/dx + 3y = 6. If y is a constant, then its derivative, dy/dx, is zero. So, the equation becomes:

0 + 3y = 6

3y = 6

y = 2

This confirms that y = 2 is indeed a solution. The initial condition y(0) = 2 simply specifies that this constant solution starts at the value 2 when x = 0.

What about the general solution? The general solution y = 2 + Ce^(-3x) represents a family of solutions. Each value of C gives us a different solution curve. As x approaches infinity, the term Ce^(-3x) approaches zero, and the solution approaches y = 2. This means that all the solutions in this family tend towards the constant solution y = 2 as x gets larger. The initial condition y(0) = 2 simply picks out the specific solution in this family where C = 0.

Conclusion

Alright guys, we've successfully solved the first-order differential equation dy/dx + 3y = 6 with the initial condition y(0) = 2. We found that the particular solution is y = 2, a constant function. We also explored the general solution and understood how the initial condition helps us pinpoint the specific solution we're looking for. This journey involved using the method of integrating factors, a powerful tool for tackling linear first-order differential equations.

Key takeaways from our adventure:

  • First-order differential equations are equations involving a function and its first derivative, used to model various real-world phenomena.
  • Linear first-order differential equations have a specific form and can be solved using an integrating factor.
  • The integrating factor transforms the equation into a form that can be easily integrated.
  • The general solution represents a family of solutions, while the particular solution satisfies a specific initial condition.
  • Initial conditions are crucial for finding the unique solution to a differential equation.

I hope this breakdown has been helpful! Differential equations can seem daunting at first, but by breaking them down into steps and understanding the underlying concepts, you can conquer them. Keep practicing, keep exploring, and keep learning! Until next time!