Understand Homogeneous Differential Equation

f(x,y)dy = g(x,y) is a differential equation of the homogeneous type, if the degree of f(x,y) and g(x,y) are the same, dx is said to be a homogeneous differential equation. A homogeneous function of degree n, for k₀, is a function of the type F(x,y) that may be expressed in the form kn F(x,y). As a result, homogeneous functions with the same degree of x and y are f and g. Changing the variable y = ux leads to an equation of the type dx/x = h(u) du, which can be easily integrated. A differential equation, on the other hand, is homogenous if it is the same function as the anonymous function and its derivatives. In linear differential equations, there are no constant terms. Integration from the solution of the homogeneous equation obtained by deleting the constant term can be used to find the solutions of any linear ordinary differential equation of any degree or order.

Homogeneous Function

A homogeneous function is one that behaves in a multiplicative scaling way. A homogeneous function is one that can be stated by writing x = kx and y = ky to construct a new function f(kx, ky) = kⁿf(x, y) where the constant k can be taken as the nth power of the exponent.

f( kx , ky) = kⁿ f( x , y )

Homogeneous Differential Equation from Homogeneous Function

A homogeneous differential equation is one that is generated using a homogeneous function. If the function f(x, y) is a homogeneous function, the differential equation dy/dx = f(x, y) is a homogeneous differential equation. Before we move on to solving a homogeneous differential equation, let’s look at another definition of a homogeneous function.

A homogeneous function of degree n is one that can be stated in the form f(x, y) = xn.g(y/x) or yn.h(x/y). We must replace y = vx and differentiate this statement y = vx with respect to x in order to solve a homogeneous differential equation of the type dy/dx = f(x, y) = g(y/x). We get dy/dx = v + x.dv/dx in this case. Let’s solve the differential equation dy/dx =f(x, y) = g(y/x) using this value of dy/dx.

g(v) = v + x.dv/dx 

g(v) – v = x.dv/dx 

dv/(g(v) – v) = dx/x

We have successfully separated the variables in this case. This can be combined with other information to discover the solution to the differential equation.

∫dv/(g(v) – v) = ∫(1/x) . dx + C

If we substitute v = y/x with this equation, we get the universal solution of the differential equation. If the homogeneous differential equation is of the form dx/dy = f(x, y), and f(x, y) is a homogeneous differential equation, we replace x = vy to discover the general solution of the differential equation dx/dy = f(x, y) = h(x/y).

Solving a Homogeneous Differential Equation in Steps

Given a type differential equation

dy/dx = F(x,y) = g(y/x)

Step 1 – In the given differential equation, substitute y = vx.

Step 2 – Upon differentiating this equation we get,

dy/dx = v + xdv/dx

Substituting the values of x and y into the provided differential equation, we get

v + xdv/dx = g(x)

xdv/dx = g(x)-v

Step 3 – Upon separating the variables, we get

dv/(g(v)-v) = dx/x

Step 4 – By integrating both sides of the equation, we arrive at

∫dv/(g(v)-v) = ∫(dx/x) + C

Step 5 – We replace v = y/x after integration.

Conclusion

In mathematics, a homogeneous function is a function of numerous variables whose value is multiplied by some power of the scalar if all of its arguments are multiplied by a scalar. This power is termed the degree of homogeneity, or simply the degree. The ordinary differential equation is converted using the substitution v=y/x.

F(x,y) (y/x) + G(x,y) = 0

into the separable differential equation, where F and G are homogeneous functions of the same degree.

xdv/dx = -[G(1,y)/F(1,y)] – v