How to solve separable differential equations

Daily Dose 037 | Mathematics

How to solve separable differential equations?

Separable differential equations.

That’s the focus for this entry, but for most, it’s a focus they’d rather pass over.

Without a doubt, the place where our students hit heavy friction for the first time in their studies is in the topic of Differential Equations.

Our mission is to help you smooth that out.

In this video, we dive into a FE Exam Practice Problem in the subject of Mathematics, specifically, dealing with how to solve Separable Differential Equations.

Key Definition

What are SEPARABLE DIFFERENTIAL EQUATIONS?

A SEPARABLE DIFFERENTIAL EQUATION is a form of nonlinear first order differential equation that can be written in the form:

• N(y)*dy/dx = M(x)

With this, we can rewrite the given SEPARABLE DIFFERENTIAL EQUATION as:

• N(y)dy = M(x)dx

This allows us to integrate both sides and move forward with defining the IMPLICIT SOLUTION.

To obtain the EXPLICIT SOLUTION we must rearrange so that the SOLUTION is in the form:

• y = y(x)

The Process

So how can I solve separable differential equation?

Though seemingly complicated, Separable Differential Equations are quite easy to solve following this process:

• Step 1: Separate the differential equation so that all of the y terms are on one side and all of the x terms are on the opposite side.

For example, say we are asked to solve the differential equation: dy/dx = (3x^2 + 2y^2)/(2xy).

Our first task would be to rearrange the equation so that all the y terms are on the left and all the x terms are on the right, such as: (2y^2 dy)/(y^2) = (3x^2 dx)/(2x)

• Step 2: Integrate both sides of the equation with respect to their variables.

Continuing the example, integrating both sides would give us: 2ln|y| = (3/2)x^2 + C

• Step 3: Solve for the constant of integration by using any known initial conditions.

In our example, let’s say we were given the initial condition of y(0) = 1.

Plugging this into our integrated equation, we’d have 2ln|1| = (3/2)(0)^2 + C and solving for C, we’d get C=0

• Step 4: Check your answer by plugging it back into the original equation derived in Step 2.

Check out this video and see how you can go about solving this type of problem in the most efficient manner while others pull out their hair trying! 🙁

As always, with Love, Prepineer

Video Review

[VIDEO] How to solve separable differential equations