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Chapter 7: Problem 5

What is a separable first-order differential equation?

Short Answer

Expert verified
Answer: To solve a separable first-order differential equation, follow these steps: 1. Identify the functions f(x) and g(y) in the given equation dy/dx = f(x)g(y). 2. Separate the variables, moving all terms involving y to the left side and all terms involving x to the right side. 3. Integrate both sides of the equation. 4. Solve for y(x) to find the general solution, depending on the specific functions f(x) and g(y).

Step by step solution

01

Definition of a Separable First-order Differential Equation

A separable first-order differential equation is a type of differential equation that can be separated into two functions, one involving only the dependent variable (usually denoted as y) and the other involving only the independent variable (usually denoted as x). The general form of a separable first-order differential equation is: dy/dx = f(x)g(y) Where f(x) is a function of x only, and g(y) is a function of y only. The goal is to solve for y(x).
02

Separating the Variables

To solve a separable first-order differential equation, we need to separate the variables, i.e., move all terms involving y to the left side and all terms involving x to the right side. This can be achieved by dividing both sides of the equation by g(y) and multiplying both sides by dx: (1/g(y)) dy = f(x) dx Now, the variables are separated, with y terms on the left and x terms on the right.
03

Integrating Both Sides

The next step is to integrate both sides, but this may not always be possible. If it is, you should obtain an equation for the general solution: 鈭�(1/g(y)) dy = 鈭玣(x) dx + C Here, C is the constant of integration.
04

Solving for y(x)

The final step is to solve for y(x) from the equation obtained in the previous step. The solution will depend on the specific functions f(x) and g(y). However, keep in mind that not all differential equations have an explicit solution. Sometimes, the solution can only be expressed implicitly.
05

Example

Let's consider an example. Suppose we have the following separable first-order differential equation: (dy/dx) = x * e^y Here, f(x) = x , and g(y) = e^y. Let's solve it step by step: 1. Separate the variables: (1/e^y) dy = x dx 2. Integrate both sides: 鈭�(1/e^y) dy = 鈭玿 dx + C Which results in: - e^{-y} = (1/2)x^2 + C 3. Solve for y(x): y(x) = -ln( -(1/2)x^2 - C ) This is the general solution for the given separable first-order differential equation.

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