Homogeneous Differential Equations

Examples of Homogeneous Differential equations

• dy/dx = x(x - y)/y²
• dy/dx = (x + y)/(x - y)
• dy/dx = (x² + y²)/xy
• dy/dx = (x³ + y³)/(xy² + yx²)

Solving a Homogeneous Differential Equation

The general solution of a homogeneous differential equation is calculated by integrating the given differential equation. 

A homogeneous differential equation: dy/dx = f(x, y) is solved by separating the variable and its derivative on either side, then integrating it with respect to the variable.  

Note: The equation for the homogeneous differential equation does not include a constant term. A constant term exists in the linear differential equation. The solution for a linear differential equation is possible if the constant term is removed from the equation and is transformed into a homogeneous differential equation.

General steps to solve a homogeneous differential equation includes:

1. We use the substitution y = v.x to solve a homogeneous differential equation of the type dy/dx = f(x, y). This replacement makes it simple to integrate and solve.
2. The differentiation of y = vx, with respect to x, gives dy/dx = v + x.dv/dx. 

3. By substituting the value of dy/dx in the expression dy/dx = f(x, y) = g(y/x) we get:

v + x.dv/dx = g(v)  ,     where v = y/x.

4. Separating the variables v and x, we get: 

5. Integrate LHS and RHS, 

6. The general solution of the given homogeneous differential equation is

To obtain the general solution of the homogeneous differential equation, we replace the value of v = y/x. The inclusion of +C in the solution indicates that it is a general solution. By solving and substituting the value of +C, we can obtain the particular solution to the given homogeneous differential equation.

Let’s understand with an example,

Q: Solve the below-mentioned differential equation.

Solution:

1. Try to make y/x terms.

2. Now we have a function of y/x.
 

3. Now use Separation of Variables:

4. Substitute v = y/x

                         Hence Solved.

Also, Try Solving:

1. dy / dx = y*( x - y ) / x²
2. dy / dx = ( x - y ) / (x + y)
3. dy / dx = ( 3x + 2y ) / 3y
4. xCos(y/x)*dy / dx = y*Cos(y/x) + x

Frequently Asked Questions

1. What is a homogeneous differential equation?
A homogeneous differential equation contains a set of variables with differentiation and a function. In a homogeneous differential equation, the function f(x, y) is a homogeneous function such that f(λx, λy) = λnf(x, y) for any non-zero constant n.

2. Why is it called a homogeneous equation?
A linear differential equation Ly = f with some function f = 0 is called a homogeneous equation because if x is a solution of Lx = 0, then kx also solves this equation for some constant k.

3. How can we prove a differential equation to be homogenous?
To prove a differential equation to be homogenous, show that it is of the form f(λx, λy) = λⁿf(x, y) for any non zero constant λ. 

Conclusion

In this article, we learned about Homogeneous differential equations. We learned that homogeneous differential equations are of the form f(λx, λy) = λnf(x, y) for any non zero constant λ. 

We also learned about the steps required to solve a homogeneous differential equation.

Now you are all set to solve any level of the homogeneous differential equation.

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