In this post, we'll learn how to solve homogeneous differential equations (DE):
What is a Homogeneous Function?
To properly understand homogeneous DEs, we'll first start by defining what homogeneous functions are.
Say we have a function
To illustrate, let's look at
First, let
Next, manipulate the function so that
The function is homogeneous if we reach the form
\) where
Homogeneous Differential Equations
If we have a DE of the form:
When we have a homogeneous DE, we can convert the form into one that is solvable using separation of variables. We can summarize the concept of this method using these steps:
- We create a placeholder expression relating
and with a random variable. - Substitute the placeholder expression to the homogeneous DE, replacing either
or . - Solve the DE with the random variable using separation of variables.
- Return the replaced
or using the placeholder expression.
Example
Consider the differential equation:
Investigate DE
First, we would investigate if this DE observes the form:
We then check if
Because both functions are of the same degree, we can use this method to solve the DE.
Apply Method
To begin the homogeneous DE solution, we relate
Afterward, we substitute the placeholder expression and its derivatives into the DE.
Next, we simplify the expression to reach a DE of the form
We can apply separation of variables afterward to reach a solution. We won't show the process, but the general solution will be:
For solutions that involve logarithms, it's better to express them in algebraic terms. We can do this by letting
From the placeholder expression
The final result is the general solution. We can verify this by evaluating the original DE with the answer.
Summary
Homogeneous differential equations are another approach to solving differential equations analytically.
Say we have a functionand a random variable . Let's evaluate it with and . The function would then be . If this expression is the same as , then we say that function is homogeneous. The variable is the degree of the homogeneous function.
If we have a DE of the form:and the functions and are homogeneous with the same degree , then we have a homogeneous differential equation.
When we have a homogeneous DE, we can convert the form into one that is solvable using separation of variables.
To use this method, we create a placeholder expression relatingand with a random variable.
Next, we substitute the placeholder expression to the homogeneous DE, replacing eitheror .
Solve the DE with the random variable using separation of variables.
Return the replacedor using the placeholder expression.