Solving The Differential Equation Xyy' = Y^2 + X√(4x^2 + Y^2) A Step-by-Step Guide

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Introduction

Differential equations, guys, can sometimes feel like navigating a maze, right? But trust me, cracking them is super rewarding! In this article, we're going to dive deep into solving a particular type of differential equation: $xyy' = y^2 + x\sqrt{4x^2 + y^2}$. This equation looks a bit intimidating at first glance, but don't worry, we'll break it down step-by-step. We will explore the techniques and substitutions required to find its solution. This is a first-order ordinary differential equation, and our goal is to find a function y(x) that satisfies this equation. To get started, we need to massage the equation into a form we can work with. Let’s roll up our sleeves and get started!

The journey of solving differential equations often involves recognizing patterns and making clever substitutions. Our main keyword here is solving this differential equation, and we want to make sure you understand each step. Remember, the key is to transform the equation into something more manageable, and we'll do just that. We'll see how a simple substitution can turn this complex-looking equation into a separable one, which is much easier to handle. We'll also discuss the importance of understanding the underlying concepts, so you're not just memorizing steps but truly grasping the method. This approach will not only help you solve this particular equation but also equip you with the tools to tackle similar problems in the future.

This article is designed to be your comprehensive guide, walking you through each step with clear explanations and insights. We will emphasize the methodical approach required for solving such equations, ensuring that you not only get the solution but also understand the why behind each step. So, whether you are a student grappling with differential equations for the first time or a seasoned mathematician looking for a refresher, this guide has something for you. Let's embark on this mathematical journey together and unravel the solution to this fascinating equation. Remember, patience and persistence are your best friends when solving differential equations. So, buckle up, and let's dive in!

Step-by-Step Solution

1. Recognizing the Homogeneous Nature

Our first key step in tackling this differential equation is to recognize that it is a homogeneous equation. But what does that mean, right? Well, in simple terms, a differential equation is homogeneous if we can rewrite it in the form y' = f(y/x). This form is crucial because it allows us to use a clever substitution that simplifies the equation. Let's check if our equation, $xyy' = y^2 + x\sqrt{4x^2 + y^2}$, fits this bill. To do this, we need to isolate y' on one side of the equation. Dividing both sides by xy, we get:

$y' = \frac{y^2 + x\sqrt{4x^2 + y^2}}{xy}$

Now, let's break this fraction apart to see if we can massage it into the desired form:

$y' = \frac{y^2}{xy} + \frac{x\sqrt{4x^2 + y^2}}{xy}$

Simplifying each term, we have:

$y' = \frac{y}{x} + \frac{\sqrt{4x^2 + y^2}}{y}$

To make the homogeneous nature even clearer, let's factor out an $x^2$ from under the square root:

$y' = \frac{y}{x} + \frac{\sqrt{x^2(4 + (y/x)^2)}}{y}$

Taking the $x$ out of the square root (assuming $x > 0$), we get:

$y' = \frac{y}{x} + \frac{x\sqrt{4 + (y/x)^2}}{y}$

Now, multiply the numerator and denominator of the second term by $x$:

$y' = \frac{y}{x} + \frac{\sqrt{4 + (y/x)^2}}{y/x}$

It's starting to look promising! Let's rewrite the first term to have a common denominator:

$y' = \frac{y/x}{1} + \frac{\sqrt{4 + (y/x)^2}}{y/x}$

Now, we can see that the entire right-hand side of the equation is expressed in terms of y/x. This confirms that our differential equation is indeed homogeneous. This recognition is a critical step because it allows us to employ the substitution method that will simplify our task considerably. Without recognizing this, we'd be stuck in the maze forever! So, let's move on to the next step, where we'll actually make the substitution and see the magic happen. Remember, the ability to identify homogeneity is a powerful tool in your differential equation-solving arsenal.

2. Applying the Substitution $v = y/x$

Now that we've confirmed our equation is homogeneous, it's time for the magic trick: the substitution! The standard substitution for homogeneous equations is $v = y/x$. This seemingly simple substitution is the key to transforming our complex equation into something much more manageable. But why does this work, you might ask? Well, by introducing v, we're essentially encapsulating the y/x terms that make the equation homogeneous. This substitution helps to reduce the complexity and structure the equation in a way that allows us to separate variables.

So, let's go ahead and apply this substitution. If $v = y/x$, then we can write $y = vx$. But we also need to express y' in terms of v and x. To do this, we'll differentiate both sides of $y = vx$ with respect to x. Using the product rule, we get:

$y' = v + x\frac{dv}{dx}$

This is a crucial piece of our puzzle! Now we have expressions for both y and y' in terms of v and x. We're ready to substitute these into our original differential equation (in its rewritten form):

$y' = \frac{y}{x} + \frac{\sqrt{4 + (y/x)^2}}{y/x}$

Substituting $y/x = v$ and $y' = v + x\frac{dv}{dx}$, we get:

$v + x\frac{dv}{dx} = v + \frac{\sqrt{4 + v^2}}{v}$

Notice anything beautiful happening here? The v terms on both sides cancel out! This is exactly what we hoped for. We're left with:

$x\frac{dv}{dx} = \frac{\sqrt{4 + v^2}}{v}$

This equation looks much simpler than our original one, doesn't it? The substitution has worked its magic, transforming a seemingly intractable equation into one that is separable. Separable equations are those where we can separate the variables (v and x in this case) onto different sides of the equation. This is a huge win because it allows us to integrate each side independently. So, the substitution $v = y/x$ was not just a random act; it was a strategic move that has brought us closer to the solution. Let's move on to the next step and actually separate these variables!

3. Separating Variables and Integrating

With the substitution $v = y/x$ done and the equation simplified to $x\frac{dv}{dx} = \frac{\sqrt{4 + v^2}}{v}$, we're now at the exciting stage of separating the variables. This is where we rearrange the equation so that all terms involving v are on one side, and all terms involving x are on the other. This separation is crucial because it allows us to integrate both sides independently, leading us closer to the solution.

To separate the variables, we'll multiply both sides of the equation by $\frac{v}{\sqrt{4 + v^2}}$ and by $\frac{dx}{x}$. This gives us:

$\frac{v}{\sqrt{4 + v^2}} dv = \frac{dx}{x}$

See how beautifully the variables are separated? All the v terms are on the left, and all the x terms are on the right. Now we can integrate both sides. Let's start with the left side. We have the integral:

$\int \frac{v}{\sqrt{4 + v^2}} dv$

This integral might look a bit tricky, but it's a classic example where a simple u-substitution comes to the rescue. Let's set $u = 4 + v^2$. Then, the derivative of u with respect to v is $\frac{du}{dv} = 2v$, or $du = 2v dv$. We can rewrite our integral in terms of u as:

$\frac{1}{2} \int \frac{1}{\sqrt{u}} du$

This integral is much easier to handle. We know that the integral of $u^{-1/2}$ is $2u^{1/2}$, so our integral becomes:

$\frac{1}{2} * 2\sqrt{u} = \sqrt{u}$

Substituting back for u, we get:

$\sqrt{4 + v^2}$

Great! We've integrated the left side. Now let's tackle the right side. We have the integral:

$\int \frac{dx}{x}$

This is a fundamental integral, and we know its result is the natural logarithm of the absolute value of x:

$\ln|x|$

Don't forget the constant of integration, C! So, combining the results of our integrations, we have:

$\sqrt{4 + v^2} = \ln|x| + C$

We've made significant progress! We've separated the variables and integrated both sides. Now we need to substitute back for v to get our equation in terms of x and y. This is the next step in our journey, and it will bring us even closer to the final solution. So, let's keep going!

4. Substituting Back and Solving for $y$

We've reached a pivotal point in solving our differential equation. We've separated the variables, integrated both sides, and now we have the equation $\sqrt{4 + v^2} = \ln|x| + C$. But remember, our original equation was in terms of x and y, and we introduced v as a substitution. So, our next crucial step is to substitute back for v to get an equation relating x and y. We know that $v = y/x$, so let's plug that back in:

$\sqrt{4 + (y/x)^2} = \ln|x| + C$

Now, we want to solve for y. This involves a bit of algebraic manipulation. First, let's square both sides of the equation to get rid of the square root:

$4 + (y/x)^2 = (\ln|x| + C)^2$

Next, we'll isolate the $(y/x)^2$ term by subtracting 4 from both sides:

$(y/x)^2 = (\ln|x| + C)^2 - 4$

Now, multiply both sides by $x^2$ to get $y^2$ on its own:

$y^2 = x^2((\ln|x| + C)^2 - 4)$

Finally, take the square root of both sides to solve for y:

$y = \pm x\sqrt{(\ln|x| + C)^2 - 4}$

And there you have it! We've found the general solution to our differential equation. This solution represents a family of curves, with each curve corresponding to a different value of the constant C. The $\pm$ sign indicates that there are two possible solutions for y for each value of x. This is a fairly complex solution, and it showcases the power of the methods we've used to arrive here. We started with a seemingly intimidating differential equation, and by recognizing its homogeneous nature, applying a clever substitution, separating variables, integrating, and substituting back, we've successfully found the solution. This journey highlights the beauty and elegance of mathematics, where complex problems can be broken down into manageable steps. Remember, the constant of integration C is crucial, as it represents the family of solutions to the differential equation. Different initial conditions would yield different values for C, thus selecting a specific solution from this family.

Conclusion

Solving the differential equation $xyy' = y^2 + x\sqrt{4x^2 + y^2}$ has been quite the adventure, hasn't it? We started with a complex-looking equation and, through a series of strategic steps, we arrived at the solution: $y = \pm x\sqrt{(\ln|x| + C)^2 - 4}$. Let's recap the key steps we took:

1. Recognizing the Homogeneous Nature: We first identified that the equation was homogeneous, meaning it could be expressed in the form y' = f(y/x). This recognition was crucial as it paved the way for our next step.
2. Applying the Substitution $v = y/x$: We made the substitution $v = y/x$, which simplified the equation considerably. This substitution transformed the equation into a separable one.
3. Separating Variables and Integrating: We separated the variables, putting all v terms on one side and all x terms on the other. Then, we integrated both sides, which involved a u-substitution for the v integral.
4. Substituting Back and Solving for $y$: Finally, we substituted back $y/x$ for v and solved for y, arriving at our general solution.

This process demonstrates the power of recognizing patterns and applying appropriate techniques in solving differential equations. The homogeneous nature of the equation was the key insight that unlocked the solution. The substitution $v = y/x$ is a standard technique for homogeneous equations, and mastering this substitution can help you solve a wide variety of problems. Separating variables and integrating are fundamental skills in differential equations, and this example provided a great opportunity to practice those skills.

Remember, guys, solving differential equations is not just about getting the right answer; it's about understanding the process. Each step we took had a purpose, and understanding why we took those steps is just as important as knowing how. This article aimed to not only provide the solution but also to explain the reasoning behind each step, so you can apply these techniques to other problems.

So, next time you encounter a differential equation, remember the strategies we used here. Look for patterns, think about substitutions, and don't be afraid to break the problem down into smaller, more manageable steps. With practice and persistence, you'll become a differential equation-solving pro! And remember, the journey of solving these equations is just as rewarding as the destination. Keep exploring, keep learning, and keep solving!