Post 3: Solving Separable Differential Equations

In our previous post, we discussed what separable differential equations are and why they are useful. Now, let's dive into the step-by-step process of solving these equations.

A separable differential equation is an equation that can be written in the form:

dy/dx = f(x)g(y)

where f(x) and g(y) are functions of x and y respectively. To solve a separable differential equation, we need to separate the variables x and y, and integrate both sides.

Step 1: Separate the variables Rearrange the equation so that all terms involving dy are on one side and all terms involving dx are on the other side. This can typically be done by multiplying both sides by a suitable function.

dy/g(y) = f(x)dx

Step 2: Integrate both sides Integrate both sides with respect to their respective variables.

∫(1/g(y))dy = ∫f(x)dx

Step 3: Solve for y Evaluate the integrals on both sides of the equation to obtain an expression for y in terms of x. This may involve using integration techniques such as substitution or partial fractions.

Step 4: Include the constant of integration Since we are integrating, it is important to include the constant of integration, denoted as C. This constant accounts for all possible solutions of the original differential equation.

Now, let's take a look at an example to illustrate these steps:

Example: Solve the separable differential equation: dy/dx = x^2y

Step 1: Separate the variables dy/y = x^2dx

Step 2: Integrate both sides ∫(1/y)dy = ∫x^2dx

Step 3: Solve for y ln|y| = (1/3)x^3 + C

Step 4: Include the constant of integration y = Ce^(x^3/3)

In this example, we successfully separated the variables, integrated both sides, and obtained the general solution for the separable differential equation.

It is important to note that there may be cases where it is not possible to solve separable differential equations analytically. In such situations, numerical approximations or software tools can be used to find approximate solutions.

Understanding how to solve separable differential equations is fundamental in calculus and has significant applications in various fields such as physics, biology, and economics. In the next post, we will explore some real-world applications of separable differential equations to further solidify our understanding.