Question

Separation of Variables In your own words, describe how to recognize and solve differential equations that can be solved by separation of variables.

Short answer

A differential equation that can be solved by the method of separation of variables can be recognized if it can be written in a form where one side is a function of \(x\) and the other is a function of \(y\). To solve it using separation of variables, terms involving \(x\) are separated from terms involving \(y\) and each side of the equation is integrated separately. Further refinement, if necessary, is done to make the solution more intelligible and the solution is checked to confirm that it matches the original equation.

Step-by-step solution

Recognition of a separable differential equation

A separable differential equation is a first order ordinary differential equation which can be written in the form of \(dy/dx = g(y) \cdot h(x)\). To recognize such an equation, look for it in the form where it can be separated into two functions, one involving only \(x\) and one involving only \(y\).

Separation of Variables

This is where the method gets its name - separation of variables. Rearrange the equation to make it in a form where one side only involves terms with \(x\) and its derivatives and the other side only involves terms with \(y\) and its derivatives. This will lead to an equation of the form \(g(y)dy = h(x)dx\). This is accomplished by simple algebraic manipulation.

Integration

After the equation is separated into two functions, each side is integrated separately. Integral of \(g(y)dy\) with respect to \(y\) is taken on one side and the integral of \(h(x)dx\) with respect to \(x\) is taken on the other side which gives solution of the differential equation. Make sure to include a constant of integration, typically denoted by \(C\), when performing indefinite integration.

Refinement and Checking of the Solution

Often, the solution obtained will need to be refined to make it more intelligible, or even to give it in an explicit form, \(y=f(x)\). This can involve tasks such as solving for \(y\) explicitly, simplifying the equation, and so on. Once the solution is obtained, it is generally a good practice to differentiate it and check if it satisfies the original differential equation, as a confirmation that the computations were performed correctly.

Key concepts

Separation of Variables

The method of separation of variables is a staple first step in solving many kinds of differential equations, particularly those that are first-order. Right off the bat, it's crucial to realize that not all differential equations are separable. A differential equation is considered separable if we can rearrange it such that all terms containing the dependent variable—typically denoted as 'y'—are on one side, and all terms containing the independent variable—'x'—are on the other side.

For example, a standard separable form looks like this: \[\frac{dy}{dx} = g(y) \cdot h(x)\]. In this situation, we can manipulate the equation to isolate terms containing 'y' and 'x' on opposite sides like so: \[g(y)dy = h(x)dx\]. What follows is integrating each side with respect to its variable, leading directly to the solutions for 'y' as a function of 'x'.

Integration of Differential Equations

Upon separating the variables, our next vital step is the integration of both sides of the equation. Integration can be seen as the reverse operation to differentiation, helping us to find the original functions from their rates of change. When you integrate \[g(y)dy\] and \[h(x)dx\], you're essentially summing up tiny changes to recover the functions 'y' and 'x'.

It's vital to remember to include the constant of integration, commonly denoted as 'C', after performing indefinite integration. This constant is very important as it accounts for the initial conditions of a problem, and without it, our solution would be incomplete. The results of these integrations give us a general solution to the differential equation.

Ordinary Differential Equations

Ordinary differential equations (ODEs) are equations that involve rates of change of a function with respect to one independent variable, and are distinguished from partial differential equations which involve multiple independent variables. Differential equations like the one in our exercise, \[dy/dx = g(y) \cdot h(x)\], are ordinary because they involve derivatives with respect to a single variable—'x' in this case.

Understanding the structure of ODEs is critical for recognizing when to use various techniques like separation of variables. Solving ODEs is fundamental in various fields such as physics, engineering, and economics because they model a myriad of real-world situations.

Algebraic Manipulation

At the heart of separating variables lies the art of algebraic manipulation. It's the transformative process we employ to rearrange an equation into a more useful or solvable form. In the context of our differential equation, algebraic manipulation is used to get all the 'y' terms on one side and the 'x' terms on the other side, allowing us to apply the separation of variables technique.

It's essential to develop a strong foundation in algebra to effectively manipulate equations. While sometimes overlooked, it’s these algebra skills that make the transition from a complex differential equation to a form where we can straightforwardly integrate each side of the equation, inching us closer to finding the solution.