How to Solve Homogeneous Differential Equations?

Homogenous differential equations are equations that contain a homogenous function. We can solve a homogeneous differential equation of the form dx/dy = f(x, y) where, f(x, y) is a homogeneous function, by simply replacing x/y to v or putting y = vx. Then after solving the differential equation, we put back the value of v to get the final solution. The detailed step for solving the Homogeneous Differential Equation i.e., dy/dx = y/x.

Step 1: Put y = vx in the given differential equation.

Now, if y = vx

then, dy/dx = v + xdv/dx

Substituting these values in the given D.E

Step 2: Simplify and then separate the independent variable and the differentiation variable on either side of the equal to.

v + xdv/dx = vx/x

⇒ v + xdv/dx = v

⇒ xdv/dx = 0

⇒ dv = 0

Step 3: Integrate the differential equation so obtained and find the general solution in v and x.

Integrating both sides,

∫dv = 0

⇒ v = c

Step 4: Put back the value of v to get the final solution in x and y.

Substituting y/x = v

⇒ y/x = c

⇒ y = cx

This is the required solution of the given homogeneous differential equation

Learn more about, How to solve Differential Equations?

Homogeneous Differential Equations are differential equations with homogenous functions. They are equations containing a differentiation operator, a function, and a set of variables. The general form of the homogeneous differential equation is f(x, y).dy + g(x, y).dx = 0, where f(x, y) and h(x, y) is a homogenous function. Homogenous functions are defined as functions in which the total power of all the terms of the function is constant. Before continuing with Homogeneous Differential Equations we should learn Homogeneous Functions first. In this article, we will learn about, Homongenous Functions, Homogeneous Differential Equations, their solutions, and others in detail.