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Homogeneous Differential Equations

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Homogeneous Differential Equations - Lesson Summary

Consider two functions F(x,y) = (2x2y2)/(x2+y2) G(x,y) = y2/(x2 - xy)

Substitute kx in place of x, and ky in place of y where k is a constant.

F(kx,ky) = (2k2x2k2y2)/(k2x2+k2y2) , G(kx,ky) = k2y2/(k2x2 - kx.ky)

F(kx,ky) = k4/k2 . (2x2y2)/(x2+y2) = k(2x2y2)/(x2+y2) ⇒ F(kx,ky) = k2F(x,y)

G(kx,ky) = k2/k2 . y2/(x2 - xy) = k0. y2/(x2 - xy) ⇒ G(kx,ky) = k0G(x,y)

This type of functions is known as homogeneous functions. Here, the function F(x, y) is of degree two, and the function G (x, y) is of degree zero.

Any homogeneous function can be expressed as F(x,y) = xng(y/x)

F(x,y) = ynh(x/y)

n = Degree of homogeneous equation

We consider the first function F(x,y) = (2x2y2)/(x2+y2)

= (2x2[x2.(y/x)2])/(x2.(1+ y2/x2)) = (2x4.(y/x)2])/(x2.(1+ y2/x2))

= (x2. 2(y/x)2])/(1+ (y/x)2) = x2F1(y/x)

This is a homogeneous function of degree two.

Similarly, G(x,y) = y2/(x2-xy)

= y2/(y2[(x/y)2 - (x/y)])  = y2-2. 1/[(x/y)2 - (x/y)] = y0 G1(x/y)

Here, zero is the degree of the function G (x, y).

A differential equation of the form dy/dx = F(x,y) is said to be homogenous if F(x,y) is a homogenous function of degree zero.

Consider a homogeneous differential equation of degree zero.

dy/dx = x0h(y/x) ................... (i)

Substitute y = vx.

dy/dx = v + x. dv/dx

equation (i) can be written as

v + x . dv/dx= x0h(v) Or v + x dv/dx = h(v)

x dv/dx = h(v) - v

⇒ 1/(h(v) - v) dv = 1/x dx

⇒ ∫ 1/(h(v) - v) dv = ∫ 1/x dx

Replace v = y/x

For the homogeneous function in the differential equation is of the form dy/dx = y0h(x/y)

Substitute x = vy in the equation.

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