Solving Dy/dx = 2yx: A Step-by-Step Guide

Hey guys! Today, we're diving into the fascinating world of differential equations. Specifically, we're going to tackle the equation dy/dx = 2yx. Don't worry if that looks intimidating – we'll break it down step by step so you can understand exactly how to find the general solution. Whether you're a student grappling with calculus or just someone curious about mathematical problem-solving, this guide is for you. We will cover everything from the basic principles to the final solution, ensuring you grasp each concept along the way. So, let's get started and unravel this mathematical puzzle together!

Understanding Differential Equations

Before we jump into solving our specific equation, let's take a moment to understand what differential equations actually are. In simple terms, a differential equation is an equation that relates a function to its derivatives. Think of it like this: you have a function, and you know something about how it changes (its rate of change, or derivative). The differential equation helps you figure out what the original function is.

Why are they important?

Differential equations are incredibly useful in many areas of science and engineering. They can model everything from the motion of objects to the flow of heat, the spread of diseases, and even the behavior of financial markets. They provide a powerful tool for understanding and predicting how systems change over time. For instance, in physics, differential equations are used to describe the motion of planets around the sun, the oscillation of a pendulum, and the behavior of electrical circuits. In biology, they can model population growth, the spread of epidemics, and the interactions between species. The versatility of differential equations makes them an essential tool in many scientific disciplines.

Types of Differential Equations

There are many different types of differential equations, but one of the most common types is the first-order differential equation. This is an equation that involves the first derivative of the unknown function. Our equation, dy/dx = 2yx, falls into this category. First-order differential equations are fundamental in various fields because they often describe the most basic relationships and changes within systems. They serve as building blocks for understanding more complex models and phenomena. Mastering first-order equations is crucial for anyone delving into mathematical modeling and analysis.

Another key distinction is between ordinary differential equations (ODEs) and partial differential equations (PDEs). ODEs involve functions of a single variable and their derivatives, while PDEs involve functions of multiple variables and their partial derivatives. Our equation is an ODE because it only involves the derivative of y with respect to x. Understanding this distinction is vital as the methods for solving ODEs and PDEs differ significantly. PDEs, for example, are used extensively in fluid dynamics, heat transfer, and electromagnetism, whereas ODEs are more commonly found in simpler systems and models.

Separating Variables: The Key Technique

Now that we have a basic understanding of differential equations, let's focus on solving our specific problem: dy/dx = 2yx. The key technique we'll use here is called separation of variables. This method is a powerful tool for solving many first-order differential equations, and it's relatively straightforward to apply.

What is Separation of Variables?

The idea behind separation of variables is to rearrange the equation so that all the terms involving the dependent variable (in our case, y) are on one side, and all the terms involving the independent variable (in our case, x) are on the other side. This makes it possible to integrate each side separately, leading us to the general solution. The method relies on the fundamental principle that if two expressions involving different variables are equal, they must each be equal to a constant. This principle allows us to split the differential equation into two simpler integrals.

Applying the Technique to dy/dx = 2yx

Let's see how this works in practice. Our equation is dy/dx = 2yx. The first step is to get all the 'y' terms on one side and all the 'x' terms on the other. To do this, we can divide both sides by 'y' and multiply both sides by 'dx'. This gives us:

(1/y) dy = 2x dx

See how we've separated the variables? All the 'y' terms are on the left, and all the 'x' terms are on the right. This separation is the critical step in solving the differential equation using this method. It allows us to treat each side independently and integrate, which we will do in the next step. The ability to separate variables is not always possible for every differential equation, but when it is, it greatly simplifies the solution process.

Integrating Both Sides

With our variables separated, the next step is to integrate both sides of the equation. This will help us move from the differential form to an algebraic equation that we can solve for y.

Integrating (1/y) dy

Let's start with the left side of our equation: (1/y) dy. The integral of 1/y with respect to y is the natural logarithm of the absolute value of y. Remember, we use the absolute value because the natural logarithm is only defined for positive values. So, we have:

∫(1/y) dy = ln|y| + C₁

Here, C₁ is the constant of integration. It's essential to include this constant because there are infinitely many functions whose derivative is 1/y, differing only by a constant term. The constant of integration captures this family of solutions.

Integrating 2x dx

Now, let's integrate the right side of our equation: 2x dx. The integral of 2x with respect to x is x². Again, we need to add a constant of integration, let's call it C₂. So, we have:

∫2x dx = x² + C₂

Similar to the left side, the constant C₂ accounts for the family of solutions that have the same derivative, 2x. Omitting the constant of integration would result in losing a significant portion of the solution set.

Combining the Results

Now that we've integrated both sides, we can combine the results. Our equation now looks like this:

ln|y| + C₁ = x² + C₂

We can simplify this a bit by combining the constants of integration into a single constant. Let's subtract C₁ from both sides:

**ln|y| = x² + (C₂ - C₁) **

We can replace (C₂ - C₁) with a new constant, let's call it C. So, our equation becomes:

ln|y| = x² + C

This equation represents the integral form of the solution. To find the explicit solution for y, we need to eliminate the natural logarithm, which we will tackle in the next step.

Solving for y

We're getting closer to our general solution! Our equation is currently in the form ln|y| = x² + C. To isolate 'y', we need to get rid of the natural logarithm. The inverse operation of the natural logarithm is the exponential function, so we'll exponentiate both sides of the equation.

Exponentiating Both Sides

Exponentiating both sides means raising 'e' (the base of the natural logarithm) to the power of each side. This gives us:

e^(ln|y|) = e^(x² + C)

By the definition of logarithms, e^(ln|y|) simplifies to |y|. On the right side, we can use the property of exponents that says e^(a+b) = e^a * e^b. So, we have:

|y| = e^(x²) * e^C

Simplifying the Constant

Since 'C' is an arbitrary constant, e^C is also an arbitrary constant. Let's replace e^C with a new constant, let's call it A. Note that A will always be a positive constant because it's the exponential of a real number. Our equation now looks like this:

|y| = A * e^(x²)

Removing the Absolute Value

To remove the absolute value, we need to consider both the positive and negative cases. If |y| = A * e^(x²), then y can be either A * e^(x²) or -A * e^(x²). We can combine these two cases by introducing another constant, let's call it B, which can be either positive or negative. So, we have:

y = B * e^(x²)

Here, B is an arbitrary constant that can be any real number (positive, negative, or zero). This is because if B is zero, y would be zero, which is also a valid solution to the original differential equation.

The General Solution

We've done it! We've successfully found the general solution to the differential equation dy/dx = 2yx. The general solution is:

y = B * e^(x²)

Where B is an arbitrary constant. This means there are infinitely many solutions to the differential equation, each corresponding to a different value of B. The constant B is determined by initial conditions, which are additional pieces of information about the function y at a particular value of x. For example, if we knew that y(0) = 5, we could substitute these values into the general solution and solve for B.

Verifying the Solution

It's always a good idea to verify our solution to make sure it's correct. To do this, we can take the derivative of our solution and see if it matches the original differential equation.

Our solution is y = B * e^(x²). Let's find its derivative with respect to x:

dy/dx = d/dx (B * e^(x²))

Using the chain rule, we get:

dy/dx = B * e^(x²) * (2x)

dy/dx = 2x * B * e^(x²)

Now, let's substitute our expression for y into the right side of the original differential equation, 2yx:

2yx = 2x * (B * e^(x²))

Notice that this is exactly the same as our expression for dy/dx! This confirms that our solution is correct. We have successfully derived the general solution to the differential equation and verified its accuracy. Great job, guys!

Conclusion

So, there you have it! We've walked through the process of finding the general solution to the differential equation dy/dx = 2yx. We started with the basics of differential equations, learned about the separation of variables technique, integrated both sides of the equation, solved for y, and verified our solution. Remember, the general solution y = B * e^(x²) represents a family of solutions, each defined by a different value of the constant B. Mastering these steps will set you up for tackling more complex problems in calculus and beyond. Keep practicing, and you'll become a differential equation whiz in no time! Happy solving!