Homogeneous Differential Equations Explained Step by Step for Beginners
Homogeneous differential equations explained step by step can make this class of first-order equations much simpler to understand. Although these equations may initially appear complicated, they become manageable once you learn how to recognize their structure and apply the appropriate substitution.
In this guide, you will learn what homogeneous differential equations are, how to identify them, and how the solution method is derived. Furthermore, we will develop a systematic procedure for solving these equations and work through several detailed examples. By the end of this tutorial, you will be able to recognize and solve homogeneous differential equations.
What Are Homogeneous Differential Equations?
Homogeneous differential equations are a special class of first-order differential equations that can be transformed into separable equations through an appropriate substitution. Their defining characteristic is that the right-hand side of the equation depends only on the ratio of the dependent variable to the independent variable.
Definition
A first-order differential equation is homogeneous if it can be written in the form
$$\frac{dy}{dx}=F(\frac{y}{x}),$$
where \( F \) is a function of \( \frac{y}{x} \).
Derivation
The solution involves transforming the equation into a separable differential equation through an appropriate substitution. Similar to the techniques used in Bernoulli Differential Equations for Beginners: Everything You Need to Know, we introduce a new variable that simplifies the equation and makes it simpler to solve. Once the equation is transformed, we can use separation of variables to solve it as discussed in the article Separable Differential Equations Explained Step by Step: A Complete Beginner’s Guide.
Consider the homogeneous differential equation
$$\frac{dy}{dx} = F(\frac{y}{x}).$$
Let
$$v = \frac{y}{x}.$$
Solving for \( y \) gives
$$y = vx.$$
Next, we differentiate the substitution. For a review of differentiation, please refer to the article How to Differentiate a Function Step by Step: A Beginner’s Guide. Using the product rule, we obtain
$$\frac{dy}{dx} = v + x\frac{dv}{dx}.$$
Substituting into the differential equation, we find
$$v + x\frac{dv}{dx} = F(v).$$
Next, subtract \( v \) from both sides to get
$$x\frac{dv}{dx} = F(v) – v.$$
Finally, dividing both sides by \( x \) we arrive at
$$\frac{dv}{dx} = \frac{F(v) – v}{x}.$$
This is a separable differential equation and can be solved using separation of variables. Don’t forget to back substitute.
This derivation shows that homogeneous differential equations can be solved using the substitution \( v= \frac{y}{x} \). The substitution transforms the original equation into a separable differential equation, enabling us to apply the method of separation of variables. In the next section, we will summarize this derivation into a step-by-step procedure that can be applied to any homogeneous differential equation.
Steps to Solve Homogeneous Differential Equations
Now that we have derived the solution method, we can develop a systematic procedure for solving homogeneous differential equations.
Step 1: Write the Equation in Standard Form
Start by writing the equation in the form
$$\frac{dy}{dx} = F(\frac{y}{x}).$$
Step 2: Introduce the Substitution \( v = \frac{y}{x} \)
Once you have verified that the equation is homogeneous, let
$$v = \frac{y}{x}.$$
Solving for \( y \) gives
$$y = vx.$$
Step 3: Differentiate \( y=vx \)
Using the product rule, we obtain
$$\frac{dy}{dx} = v + x\frac{dv}{dx}.$$
Step 4: Substitute into the Original Equation
Replace
$$\frac{dy}{dx}$$
with
$$v + x\frac{dv}{dx},$$
and replace every occurrence of
$$\frac{y}{x}$$
with
$$v.$$
After substitution, the equation should involve only the variables \( v \) and \( x \).
Step 5: Rearrange the Equation
In general, the equation will take the form
$$\frac{dv}{dx} = \frac{F(v) – v}{x},$$
Step 6: Solve the Resulting Separable Differential Equation
Since the equation is now separable, separation of variables can be used.
Step 7: Convert Back to the Original Variable
Back substituting solves the original homogeneous differential equation.
Step 8: Apply the Initial Condition (If Given)
If an initial condition is provided, substitute it into the general solution to determine the constant C.
Worked Out Example
The best way to understand homogeneous differential equations is to solve many examples. In the example below, we will follow the procedure developed in the previous sections.. This example also requires the techniques covered in the articles Basic Integration Problems for Beginners and The Ultimate Step by Step Guide to Solving Integrals Using Substitution.
Example 1: Solve \( \frac{dy}{dx} = \frac{x + y}{x – y} \).
Solution: Multiplying the numerator and denominator by \( \frac{1}{x} \) gives
$$\frac{dy}{dx} = \frac{1 + \frac{y}{x}}{1 – \frac{y}{x}}.$$
Let
$$v = \frac{y}{x}.$$
This implies
$$y=vx.$$
Then
$$\frac{dy}{dx} = v + x\frac{dv}{dx}.$$
Substituting into the differential equation, we obtain
$$v + x\frac{dv}{dx} = \frac{1 + v}{1 – v}.$$
Subtracting \( v \) from both sides we get
$$x\frac{dv}{dx} = \frac{1 + v}{1 – v} – v.$$
Multiplying both sides by \( \frac{1}{x} \) this is
$$\frac{dv}{dx} = (\frac{1 + v}{1 – v} – v)\frac{1}{x}.$$
Separating variables, we have
$$\frac{1}{\frac{1 + v}{1 – v} – v} dv = \frac{1}{x} dx.$$
Integrating both sides, we find
$$\int \frac{1}{\frac{1 + v}{1 – v} – v} dv = \int \frac{1}{x} dx.$$
To evaluate the integral on the left-hand side, multiply the numerator and denominator by \( 1 – v \) to get
$$\int \frac{1 – v}{1 + v – v(1 – v)} dv = \int \frac{1}{x} dx.$$
Distributing we obtain
$$\int \frac{1 – v}{1 + v – v + v^2} dv = \int \frac{1}{x} dx.$$
Which is equivalent to
$$\int \frac{1 – v}{1 + v^2} dv = \int \frac{1}{x} dx.$$
Splitting the fraction on the left into two gives
$$\int \frac{1}{1 + v^2} – \frac{v}{1 + v^2} dv = \int \frac{1}{x} dx.$$
Therefore
$$\arctan{v} – \frac{1}{2}(\ln|1 + v^2| = \ln|x| + C.$$
Our final solution is then
$$\arctan(\frac{y}{x}) – \frac{1}{2}(\ln|1 + (\frac{y}{x})^2| = \ln|x| + C.$$
Conclusion
In this guide on homogeneous differential equations explained, we saw how a homogeneous differential equation can be transformed into a separable differential equation, allowing us to apply separation of variables to obtain the solution. We also developed a step-by-step procedure for solving homogeneous differential equations and applied it to several worked examples.
Further Reading
How to Solve Exact Differential Equations: A Beginner-Friendly Tutorial with Worked Examples – Exact differential equations are another one of the basic differential equations you should be able to solve. This article covers these equations in depth.
