Solving (x²+y²+3)dy/dx = 2x(2y-x²/y): A Step-by-Step Guide

Oct 19, 2025 by Blender 59 views

Solving the Differential Equation (x²+y²+3)dy/dx = 2x(2y-x²/y): A Comprehensive Guide

Hey guys! Today, we're diving deep into solving a fascinating differential equation: (x²+y²+3)dy/dx = 2x(2y-x²/y). If you're scratching your head trying to figure this out, don't worry! We'll break it down step-by-step, making it super clear and easy to understand. So, buckle up and let's get started!

Understanding the Problem

Before we jump into the solution, let's really get what this equation is all about. This is a first-order ordinary differential equation (ODE). First-order means that the highest derivative involved is the first derivative, dy/dx. Ordinary means that we're dealing with a function of a single independent variable (x in this case). Our goal is to find a function y(x) that satisfies this equation.

At first glance, this equation might look intimidating. It's not immediately obvious what method we should use to solve it. It's definitely not a simple separable equation, and it doesn't quite fit the mold of a linear first-order ODE either. That's where the fun begins! We'll need to use some algebraic manipulation and a clever substitution to transform it into a form we can handle. Think of it like a puzzle – we need to rearrange the pieces until the solution becomes clear.

Differential equations are a cornerstone of many fields, including physics, engineering, and economics. They describe how things change, from the motion of a pendulum to the growth of a population. Mastering these equations is crucial for anyone working in these areas. So, by tackling this problem, we're not just solving a math equation; we're honing skills that have real-world applications. In the following sections, we'll explore the techniques needed to solve this equation and similar ones. Remember, the key is to break down the problem into smaller, manageable steps and to understand the underlying concepts. So, let's dive into the first step of our solution!

Step 1: Simplifying the Equation

Okay, so let's tackle this equation head-on. The first thing we want to do is simplify the equation to make it easier to work with. We can start by getting rid of that fraction inside the parenthesis on the right side. Let's multiply both sides of the equation by 'y' to clear the fraction:

y(x² + y² + 3) dy/dx = 2x(2y² - x²)

Now, let's expand the right side:

y(x² + y² + 3) dy/dx = 4xy² - 2x³

At this point, it might not be immediately obvious where to go next, and that's perfectly fine! Math often involves a bit of exploration. The key is to keep trying different approaches until something clicks. One thing we might notice is the presence of x², y², and xy terms. These often hint at the possibility of using a substitution to simplify the equation. Specifically, we might think about substitutions that relate y and x, such as y = vx. This is a common technique for dealing with homogeneous equations, and while our equation isn't quite homogeneous in its current form, this substitution could still be helpful.

Another thing to consider is whether this equation might be exact. An exact differential equation is one that can be written in the form M(x, y) dx + N(x, y) dy = 0, where ∂M/∂y = ∂N/∂x. If our equation is exact, we can solve it by finding a potential function. To check if our equation is exact, we'll need to rearrange it into the standard form and then compute the partial derivatives. This is another avenue we can explore, and it's worth keeping in mind as we proceed.

For now, let's focus on the substitution y = vx. This is a powerful technique that can often simplify complex differential equations. In the next step, we'll see how this substitution transforms our equation and brings us closer to the solution. Remember, the goal here is to make the equation more manageable, and substitutions are a great way to achieve that. Keep experimenting and exploring, and you'll be amazed at what you can discover!

Step 2: Applying the Substitution y = vx

Alright, let's try the substitution y = vx. This is a classic move for tackling equations with terms like x² and y². The idea here is to introduce a new function, v(x), that relates y and x. This substitution can often simplify the equation by transforming it into a more manageable form. But remember, if y = vx, then we also need to find an expression for dy/dx. To do this, we'll use the product rule:

dy/dx = v + x(dv/dx)

Now we have everything we need to substitute into our original simplified equation: y(x² + y² + 3) dy/dx = 4xy² - 2x³. Let's replace y with vx and dy/dx with v + x(dv/dx):

vx(x² + (vx)² + 3)(v + x(dv/dx)) = 4x(vx)² - 2x³

This might look even more complicated at first, but trust me, we're making progress! Now, let's simplify this beast. First, we can expand the terms and see if anything cancels out. We have:

vx(x² + v²x² + 3)(v + x(dv/dx)) = 4vx³ - 2x³

Next, let's divide both sides by x³ (assuming x ≠ 0): Note : It is essential to consider the situation when x = 0 later in your solution

v(1 + v² + 3/x²)(v + x(dv/dx)) = 4v² - 2

This is starting to look a bit cleaner. We've managed to eliminate some of the x terms, which is a good sign. The equation now involves v, dv/dx, and x, which means we're on the right track to potentially separating variables. The substitution y = vx has helped us transform the equation into a form where we can start to isolate the variables and integrate. In the next step, we'll continue simplifying and try to separate the variables so we can finally integrate and solve for v(x).

Step 3: Separating Variables

Okay, guys, let's get those variables separated! Remember our equation from the last step?

v(1 + v² + 3/x²)(v + x(dv/dx)) = 4v² - 2

This looks a bit tangled, but we can definitely work with it. Let's start by expanding the left side:

v(1 + v² + 3/x²)(v + x(dv/dx)) = v³ + vx(dv/dx) + v⁵ + v³x(dv/dx) + (3v/x²)v + (3v/x²)x(dv/dx)

Simplifying, we get:

v³ + vx(dv/dx) + v⁵ + v³x(dv/dx) + 3v²/x² + (3v/x)(dv/dx) = 4v² - 2

Now, let's group the terms with dv/dx:

[vx + v³x + (3v/x)] dv/dx = 4v² - 2 - v³ - v⁵ - 3v²/x²

This is still quite a handful, but we're getting closer. The goal here is to get all the terms involving 'v' on one side and all the terms involving 'x' on the other side. This is the essence of separating variables.

To make things clearer, let's factor out an 'x' from the dv/dx terms on the left and find a common denominator on the right:

x[v + v³ + (3v/x²)] dv/dx = (4v² - 2 - v³ - v⁵ - 3v²/x²)

x[v + v³ + (3v/x²)] dv/dx = (-v⁵ - v³ + 4v² - 2 - 3v²/x²)

It seems like we made a slight detour that complicated things further. Let's backtrack to v(1 + v² + 3/x²)(v + x(dv/dx)) = 4v² - 2 and try a different approach to isolate dv/dx:

v(1 + v² + 3/x²)(v + x dv/dx) = 4v² - 2

Divide both sides by v(1 + v² + 3/x²) :

v + x dv/dx = (4v² - 2) / (v(1 + v² + 3/x²))

Now, subtract 'v' from both sides:

x dv/dx = (4v² - 2) / (v(1 + v² + 3/x²)) - v

Find a common denominator on the right:

x dv/dx = [(4v² - 2) - v²(1 + v² + 3/x²)] / [v(1 + v² + 3/x²)]

Simplify the numerator:

x dv/dx = [4v² - 2 - v² - v⁴ - 3v²/x²] / [v(1 + v² + 3/x²)]

x dv/dx = [-v⁴ + 3v² - 2 - 3v²/x²] / [v(1 + v² + 3/x²)]

Now we can invert both sides to get dx and dv on opposite sides:

(x)⁻¹ dx = [v(1 + v² + 3/x²)] / [-v⁴ + 3v² - 2 - 3v²/x²] dv

Multiply both sides by dx and by the reciprocal of the term multiplied by dv:

dx/x = [v(1 + v² + 3/x²)] / [-v⁴ + 3v² - 2 - 3v²/x²] dv

Multiply the numerator and denominator on the right side by x² to get rid of the fraction within the fraction:

dx/x = [v(x² + v²x² + 3)] / [-x²v⁴ + 3x²v² - 2x² - 3v²] dv

This looks like a separated equation! We have all the 'x' terms on the left and all the 'v' terms on the right. However, the right side is still quite complex, and we'll need to use some clever integration techniques to solve it.

Step 4: Integration (The Tricky Part!)

Okay, guys, we've reached the integration step, which, let's be honest, is often the trickiest part of solving differential equations. We've successfully separated our variables, and we now have:

dx/x = [v(x² + v²x² + 3)] / [-x²v⁴ + 3x²v² - 2x² - 3v²] dv

Let's rewrite this in the integral form:

∫ dx/x = ∫ [v(x² + v²x² + 3)] / [-x²v⁴ + 3x²v² - 2x² - 3v²] dv

The left side is straightforward: ∫ dx/x = ln|x| + C₁. But that right side… wowza! It's a beast. This is where things get interesting and where we might need to pull out some advanced integration techniques. Polynomial long division, partial fraction decomposition, or even a trigonometric substitution could be in order. But let's hold our horses for a second. Notice the x² terms on the right side. Remember, that the variables are separated, so each side of the equation is dependent on its own variable.

This suggests that our separation of variables might not be correct, or this could be a very difficult integral. Let's go back and re-examine our previous steps to see if we can find a cleaner way to separate the variables or simplify the integral. (This is the reality of solving differential equations – sometimes you need to backtrack and try a different approach!)

Going back to the equation before multiplying by x²:

dx/x = [v(1 + v² + 3/x²)] / [-v⁴ + 3v² - 2 - 3v²/x²] dv

Let's multiply numerator and denominator by x²:

dx/x = [v(x²(1 + v²) + 3)] / [-x²(v⁴ - 3v² + 2) - 3v²] dv

This didn't really help, and we still have the x² term inside the integral. It seems like this path will lead to a very complicated integral. Maybe there's a better substitution or a different approach we can take from the beginning.

(At this point, in a real problem-solving scenario, you might consider using a symbolic math software like Mathematica or Maple to help with the integration. However, for the sake of learning, we'll try to find an analytical solution.)

Let's revisit our original equation after the substitution y = vx but before we separated the variables:

v(1 + v² + 3/x²)(v + x dv/dx) = 4v² - 2

Instead of isolating dv/dx directly, let's try expanding the left side more carefully and see if any cancellations occur that we missed:

v(1 + v² + 3/x²)(v + x dv/dx) = v³ + vx dv/dx + v⁵ + v³x dv/dx + 3v/x² * v + 3v/x² * x dv/dx

Simplify:

v³ + vx dv/dx + v⁵ + v³x dv/dx + 3v²/x² + 3v/x dv/dx = 4v² - 2

Group the dv/dx terms:

x(v + v³ + 3v/x²) dv/dx = 4v² - 2 - v³ - v⁵ - 3v²/x²

This still looks messy, but let's try another manipulation. Multiply both sides of the original equation v(1 + v² + 3/x²)(v + x dv/dx) = 4v² - 2 by x² to clear the fraction inside the parenthesis:

v(x² + v²x² + 3)(v + x dv/dx) = x²(4v² - 2)

Expand:

(vx² + v³x² + 3v)(v + x dv/dx) = 4x²v² - 2x²

v²x² + vx³ dv/dx + v⁴x² + v³x³ dv/dx + 3v² + 3vx dv/dx = 4x²v² - 2x²

Group dv/dx terms:

x(vx² + v³x² + 3v) dv/dx = 4x²v² - 2x² - v²x² - v⁴x² - 3v²

Simplify:

xv(x² + v²x² + 3) dv/dx = x²(4v² - 2 - v² - v⁴) - 3v²

xv(x² + v²x² + 3) dv/dx = x²(3v² - v⁴ - 2) - 3v²

Still not quite there, but we're exploring different avenues. It's possible that this equation requires a more advanced technique or a numerical solution.

Let's pause here and recap what we've tried:

Simplified the original equation.
Applied the substitution y = vx and found dy/dx.
Substituted into the equation and tried to separate variables.
Encountered a complex integral and realized we needed to rethink our approach.

We've explored several paths, and while we haven't reached a complete solution yet, we've learned a lot about the challenges involved in solving differential equations. Sometimes, the most valuable lesson is knowing when to step back and try a different strategy. In the next step, we'll explore another possible approach or consider the limitations of analytical solutions for this particular equation.

Step 5: Exploring Alternative Approaches and Conclusion

Okay, guys, we've given this problem a good run for its money, and we've hit a bit of a roadblock with direct integration. That's perfectly okay! In the world of differential equations, not everything has a neat and tidy analytical solution. Sometimes, the best approach is to explore alternative methods or consider numerical solutions.

So, let's recap where we are. We've simplified the equation, applied the substitution y = vx, and attempted to separate variables. We ran into a very complex integral that doesn't seem to have an elementary solution. This suggests that our initial approach might not be the most fruitful one.

Here are a few alternative avenues we could explore:

Checking for Exactness: We briefly mentioned this earlier, but it's worth revisiting. We can rearrange the equation into the form M(x, y) dx + N(x, y) dy = 0 and check if ∂M/∂y = ∂N/∂x. If it's exact, we can find a potential function and solve the equation.
Numerical Methods: If we can't find an analytical solution, we can use numerical methods to approximate the solution. Methods like Euler's method or Runge-Kutta methods can give us a good approximation of the solution curve.
Graphical Analysis: We can use software to plot the direction field of the differential equation. This can give us a visual understanding of the behavior of the solutions.
Lie Group Methods: This is a more advanced technique, but it can sometimes be used to find symmetries in the differential equation, which can lead to a solution.

Unfortunately, fully exploring each of these methods in detail would make this guide incredibly long. However, the key takeaway here is that when one approach hits a wall, it's time to broaden our horizons and consider other techniques.

Conclusion:

While we haven't arrived at a closed-form solution for the differential equation (x²+y²+3)dy/dx = 2x(2y-x²/y) in this guide, we've walked through a comprehensive problem-solving process. We've explored simplification techniques, substitutions, and the challenges of integration. We've also highlighted the importance of recognizing when to try alternative methods and the existence of numerical and graphical approaches.

Differential equations can be tough nuts to crack, and this particular equation seems to be one of the tougher ones! But the journey is just as important as the destination. By working through this problem, we've honed our skills, expanded our knowledge, and gained a deeper appreciation for the beauty and complexity of differential equations. Keep practicing, keep exploring, and you'll become a master problem-solver in no time!

Remember, guys, math is all about the journey, not just the final answer. So, keep exploring, keep learning, and keep those questions coming! You've got this!