Solving Dy/dx = 9xe^y: A Step-by-Step Guide

Are you grappling with differential equations and initial conditions? Look no further! This comprehensive guide will walk you through the process of solving the differential equation dy/dx = 9xe^y with the initial condition y(0) = 0. We'll break down each step, ensuring you understand the underlying concepts and can confidently tackle similar problems. Let's dive in!

1. Introduction to Differential Equations

Before we jump into the solution, let's briefly discuss what differential equations are and why they're important. In essence, a differential equation is an equation that relates a function with its derivatives. These equations are fundamental in modeling various phenomena in science and engineering, from the motion of objects to the flow of heat. The equation we're tackling, dy/dx = 9xe^y, is a first-order differential equation because it involves only the first derivative of the function y with respect to x.

Differential equations are equations that involve an unknown function and its derivatives. They are crucial for modeling real-world phenomena in physics, engineering, biology, and economics. Solving a differential equation means finding the function that satisfies the equation. Our main keywords here are differential equations, which are mathematical equations that relate a function with its derivatives. These equations are indispensable tools for modeling a wide range of phenomena in diverse fields such as physics, engineering, biology, and economics. The central task in dealing with differential equations is to find the function that satisfies the given equation, which often involves techniques like separation of variables or using integrating factors. Understanding differential equations is essential for anyone delving into advanced mathematical modeling and problem-solving across various scientific and engineering disciplines. So, mastering these concepts can open doors to numerous exciting applications and research opportunities.

Initial conditions play a vital role in finding a unique solution to a differential equation. Think of a differential equation as describing a family of curves. An initial condition acts as a specific point that the solution curve must pass through, thus pinpointing one particular curve from the entire family. In our case, the initial condition is y(0) = 0, which means the solution curve must pass through the point (0, 0) on the Cartesian plane. This condition will be crucial later when we determine the constant of integration. Without an initial condition, we'd only have a general solution, representing a whole set of possible solutions rather than a single, definitive one. Initial conditions are the linchpin that allows us to nail down a specific solution that accurately models the situation at hand.

2. Separating Variables: The Key to Solving

The given differential equation, dy/dx = 9xe^y, is a separable equation. This means we can rearrange the equation so that all the terms involving y are on one side and all the terms involving x are on the other side. This is a crucial step in solving many first-order differential equations. To separate the variables, we'll divide both sides by e^y and multiply both sides by dx. This gives us:

e^(-y) dy = 9x dx

Now, all the y terms are on the left side, and all the x terms are on the right side. This separation allows us to integrate both sides independently, a cornerstone technique in solving such equations. Separation of variables is a fundamental approach because it transforms a complex differential equation into two simpler integrals, making the solution process much more manageable. When you come across a differential equation, always check if it's separable—it's often the quickest route to the solution.

The separation of variables technique is a cornerstone in solving first-order differential equations, and it’s precisely what we’ll use to tackle dy/dx = 9xe^y. The beauty of this method lies in its simplicity and effectiveness. By skillfully rearranging the equation, we isolate the terms involving y on one side and the terms involving x on the other. Think of it as sorting puzzle pieces so you can work on each section separately. This process transforms a single differential equation into two integrals, each of which is often easier to handle. For our equation, the separation process involves dividing both sides by e^y and multiplying both sides by dx, leading us to the pivotal form e^(-y) dy = 9x dx. This is more than just algebraic manipulation; it's a strategic move that sets the stage for integration, paving the way toward finding the solution. Mastering this technique is like adding a key tool to your problem-solving arsenal in calculus and beyond.

3. Integrating Both Sides: Finding the General Solution

With the variables separated, we can now integrate both sides of the equation. Integrating e^(-y) dy with respect to y gives us -e^(-y). Integrating 9x dx with respect to x gives us (9/2)x^2. Don't forget to add the constant of integration, which we'll call C. So, we have:

-e^(-y) = (9/2)x^2 + C

This equation represents the general solution to the differential equation. It's a family of solutions, each corresponding to a different value of C. The constant of integration arises because the derivative of a constant is zero, meaning there are infinitely many functions that could have the same derivative. To find the specific solution that satisfies our initial condition, we need to determine the value of C.

Integration is the inverse operation of differentiation, and it’s a fundamental step in solving differential equations. When we integrate both sides of the separated equation, we're essentially undoing the differentiation process to reveal the original function. The integral of e^(-y) with respect to y is -e^(-y), and the integral of 9x with respect to x is (9/2)x^2. Always remember to add the constant of integration, C, to one side of the equation. This constant accounts for the fact that the derivative of a constant is zero, meaning there are infinitely many functions that could have the same derivative. Our integration step yields -e^(-y) = (9/2)x^2 + C, which represents the general solution to the differential equation. This general solution is like a family of curves, each corresponding to a different value of C. To pinpoint the specific solution we need, we'll use the initial condition in the next step, effectively selecting one particular curve from this family. Integrating accurately and understanding the significance of the constant of integration are crucial skills in solving differential equations.

4. Applying the Initial Condition: Finding the Particular Solution

This is where the initial condition, y(0) = 0, comes into play. This condition tells us that when x = 0, y = 0. We can substitute these values into the general solution to solve for C:

-e^(-0) = (9/2)(0)^2 + C

Simplifying this, we get:

-1 = 0 + C

Therefore, C = -1. Now we can substitute this value back into the general solution to get the particular solution:

-e^(-y) = (9/2)x^2 - 1

The initial condition is the key that unlocks the specific solution to our differential equation. Think of the general solution as a map of possibilities, while the initial condition acts as a pinpoint, directing us to the exact location we need. In our case, the initial condition y(0) = 0 tells us that when x is 0, y is also 0. We substitute these values into the general solution, -e^(-y) = (9/2)x^2 + C, to solve for the constant of integration, C. This step is crucial because it transforms a family of solutions into a single, unique solution that fits the specific scenario we're modeling. After plugging in our values, we find that C = -1. This is the missing piece that allows us to write the particular solution: -e^(-y) = (9/2)x^2 - 1. This equation represents the one specific curve that satisfies both the differential equation and the initial condition. Without the initial condition, we'd be left with a range of possible solutions, making it an indispensable part of solving many differential equation problems.

5. Solving for y: Isolating the Dependent Variable

To express the solution explicitly in terms of y, we need to isolate y. First, multiply both sides of the equation by -1:

e^(-y) = 1 - (9/2)x^2

Next, take the natural logarithm of both sides:

-y = ln(1 - (9/2)x^2)

Finally, multiply both sides by -1 to solve for y:

y = -ln(1 - (9/2)x^2)

This is the explicit solution to the differential equation that satisfies the given initial condition.

Isolating the dependent variable, in our case y, is the final step in expressing the solution in a clear and usable form. We've found the relationship between x and y, but to truly understand how y behaves as x changes, we need to get y by itself. This process involves a series of algebraic manipulations. First, we multiply both sides of the equation -e^(-y) = (9/2)x^2 - 1 by -1 to get e^(-y) = 1 - (9/2)x^2. Then, we take the natural logarithm of both sides, which allows us to bring down the -y exponent, giving us -y = ln(1 - (9/2)x^2). Finally, we multiply both sides by -1 to solve for y: y = -ln(1 - (9/2)x^2). This is our explicit solution, and it tells us exactly how y depends on x. It's a critical step because it transforms an implicit solution, where y is mixed up with other terms, into an explicit one, where y is expressed directly as a function of x. This final form is much easier to analyze, graph, and use in further calculations.

6. Verifying the Solution: A Crucial Check

It's always a good idea to verify your solution to ensure it's correct. To do this, we can differentiate the solution we found, y = -ln(1 - (9/2)x^2), with respect to x and see if we get back the original differential equation.

First, find the derivative of y with respect to x:

dy/dx = -[1 / (1 - (9/2)x^2)] * (-9x)

Simplifying this, we get:

dy/dx = 9x / (1 - (9/2)x^2)

Now, we need to show that this is equal to 9xe^y. Recall that e^y = 1 / (1 - (9/2)x^2). Substituting this into the right side of the original differential equation, we get:

9xe^y = 9x * [1 / (1 - (9/2)x^2)]

This is the same as the derivative we found, so our solution is verified. This step confirms that our solution not only satisfies the initial condition but also correctly represents the relationship described by the original differential equation.

Verification is a pivotal step in solving differential equations, acting as a safeguard against potential errors. Think of it as the final quality check before you declare victory. The process involves taking the solution we've obtained and plugging it back into the original differential equation to ensure it satisfies the equation. In our case, we differentiate y = -ln(1 - (9/2)x^2) with respect to x and obtain dy/dx = 9x / (1 - (9/2)x^2). To verify this, we also need to show that it's equivalent to 9xe^y. Since we know y = -ln(1 - (9/2)x^2), we can express e^y as 1 / (1 - (9/2)x^2). Substituting this into 9xe^y, we get 9x * [1 / (1 - (9/2)x^2)], which is indeed the same as our calculated dy/dx. This confirms that our solution is correct. Verification not only provides assurance in the correctness of the solution but also deepens understanding by reinforcing the connection between the solution and the original equation. It’s a practice that should always be a part of your problem-solving routine.

7. Conclusion

In this guide, we've walked through the process of solving the differential equation dy/dx = 9xe^y with the initial condition y(0) = 0. We covered separating variables, integrating both sides, applying the initial condition, solving for y, and verifying the solution. By understanding these steps, you'll be well-equipped to tackle a wide range of differential equation problems. Remember to practice regularly and don't hesitate to revisit these steps as needed. Happy solving!

For further learning and practice on differential equations, you can explore resources like Khan Academy's Differential Equations Course. This trusted website offers comprehensive lessons and exercises to enhance your understanding and skills in this area.