Question

The differential equation \(y^{\prime}=x-y\) is separable.

Short answer

The general solution is \( y = x - C_1 e^{-x} \).

Step-by-step solution

Identify the Structure

The differential equation given is \( y' = x - y \). This equation is indeed separable because it can be written in a form where we can express all \( y \)'s and derivatives of \( y \) on one side and all \( x \)'s on the other.

Rearrange the Equation

To separate the variables, we first rewrite the equation as \( y' = x - y \). This can be converted to \( y' + y = x \). We aim to express the equation in a form suitable for separation.

Separate Variables

Now express the equality in the form \( y' = x - y \) as \( dy/dx = x - y \). To separate variables, rewrite it as \( dy/(x-y) = dx \). This ensures that all \( y \)'s are on one side and all \( x \)'s are on the other.

Integrate Both Sides

Integrate both sides of the equation. The left side becomes \( \int \frac{1}{x-y} \, dy \), and the right side becomes \( \, \int \, dx \). Solving these integrals will help us find a general solution.

Solve the Integrals

The integral on the left is \( -\ln|x-y| \), and on the right, it is \( x + C \), where \( C \) is a constant of integration. Therefore, we have \( -\ln|x-y| = x + C \).

Solve for y

Rearrange \( -\ln|x-y| = x + C \) to solve for \( y \). Raise both sides to the power of \( e \) to eliminate the logarithm: \( |x-y| = e^{-x-C} \). Let \( e^{-C} = C_1 \), a new constant, then \( x-y = C_1 e^{-x} \). Therefore, \( y = x - C_1 e^{-x} \).

Formulate Final Solution

The general solution to the differential equation is \( y = x - C_1 e^{-x} \), where \( C_1 \) is any constant.

Key concepts

Variable Separation

Separable differential equations allow us to separate the variables involved in the equation, isolating one variable and its derivatives on one side and the other variable on the other side. This is crucial because it simplifies the process of solving the equation by integration. In the given problem, our differential equation is initially stated as \( y' = x - y \). Here, we seek to separate the variables, enabling each side of the equation to involve only one variable.
We rearrange this expression to put it into a form suitable for separation: \( y' = x - y \) becomes \( y' + y = x \), then transforms into \( dy = (x - y)dx \).
Finally, we achieve the separated form by expressing it as \( \frac{dy}{x-y} = dx \), thus preparing it for the next step, which is integration.

Integration Techniques

Integration is the technique used to solve separable differential equations after the variables have been separated. With the variables in separate forms, we now need to integrate each side. In our equation, we have: \( \int \frac{dy}{x-y} = \int dx \).
The integral on the right side \( \int dx \) simply gives us \( x + C \), where \( C \) is the constant of integration.
For the integral of the left side \( \int \frac{dy}{x-y} \), it shifts to focus on the relationship \( -\ln|x-y| \). This result uses the technique related to the direct integration of functions of the form \( \frac{1}{u} \), yielding a natural logarithm.
Thus, integrating both sides provides \( -\ln|x-y| = x + C \), pointing us towards finding the general solution.

General Solution

The general solution of a differential equation is obtained after integrating and rearranging the variables to solve for the dependent variable. Back in our main problem, once the separated equation \( -\ln|x-y| = x + C \) is integrated, the next step is simplifying and solving it to express \( y \) as a function in terms of \( x \).
To dissolve the equation further and "free" \( y \) from the logarithm, we take the exponential of both sides: \( |x-y| = e^{-x-C} \).
Introducing a new constant \( C_1 = e^{-C} \), we rearrange it to obtain: \( x-y = C_1 e^{-x} \), and thus, the function of \( y \) becomes clearly defined as: \( y = x - C_1 e^{-x} \).
This expression represents the general solution that applies to any specific solution upon assigning values to \( C_1 \).

Constant of Integration

The constant of integration represents an essential part of the general solution to a differential equation, signifying an entire family of potential specific solutions. As we solve our differential equation, \( -\ln|x-y| = x + C \), the \( C \) represents this constant of integration.
When integrating indefinitely, this constant arises because any derivative of a constant term is zero, meaning multiple functions can have the same derivative.
Upon solving the function further, it was represented as \( e^{-C} = C_1 \), converting \( C \) into a multipliable constant within \( e^{-x-C} \), encapsulating all possibilities into \( C_1 \). Hence, the comprehensive solution appears as \( y = x - C_1 e^{-x} \), where \( C_1 \) can be any real number, determining how the solution fits specific conditions or initial values.