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Consider two functions F(x,y) = (2x^{2}y^{2})/(x^{2}+y^{2}) G(x,y) = y^{2}/(x^{2} - xy)

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

F(kx,ky) = (2k^{2}x^{2}k^{2}y^{2})/(k^{2}x^{2}+k^{2}y^{2}) , G(kx,ky) = k^{2}y^{2}/(k^{2}x^{2} - kx.ky)

F(kx,ky) = k^{4}/k^{2} . (2x^{2}y^{2})/(x^{2}+y^{2}) = k^{2 }(2x^{2}y^{2})/(x^{2}+y^{2}) â‡’ F(kx,ky) = k^{2}F(x,y)

G(kx,ky) = k^{2}/k^{2} . y^{2}/(x^{2} - xy) = k^{0}. y^{2}/(x^{2} - xy) â‡’ G(kx,ky) = k^{0}G(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) = x^{n}g(y/x)

F(x,y) = y^{n}h(x/y)

n = Degree of homogeneous equation

We consider the first function F(x,y) = (2x^{2}y^{2})/(x^{2}+y^{2})

= (2x^{2}[x^{2}.(y/x)^{2}])/(x^{2}.(1+ y^{2}/x^{2})) = (2x^{4}.(y/x)^{2}])/(x^{2}.(1+ y^{2}/x^{2}))

= (x^{2}. 2(y/x)^{2}])/(1+ (y/x)^{2}) = x^{2}F_{1}(y/x)

This is a homogeneous function of degree two.

Similarly, G(x,y) = y^{2}/(x^{2}-xy)

= y^{2}/(y^{2}[(x/y)^{2} - (x/y)]) = y^{2-2}. 1/[(x/y)^{2} - (x/y)] = y^{0} G_{1}(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 = x^{0}h(y/x) ................... (i)

Substitute y = vx.

dy/dx = v + x. dv/dx

equation (i) can be written as

v + x . dv/dx= x^{0}h(v) Or v + x dv/dx = h(v)

x dv/dx = h(v) - v

â‡’ 1/(h(v) - v) dv = 1/x dx

â‡’ âˆ« 1/(h(v) - v) dv = âˆ« 1/x dx

Replace v = y/x

For the homogeneous function in the differential equation is of the form dy/dx = y^{0}h(x/y)

Substitute x = vy in the equation.

Consider two functions F(x,y) = (2x^{2}y^{2})/(x^{2}+y^{2}) G(x,y) = y^{2}/(x^{2} - xy)

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

F(kx,ky) = (2k^{2}x^{2}k^{2}y^{2})/(k^{2}x^{2}+k^{2}y^{2}) , G(kx,ky) = k^{2}y^{2}/(k^{2}x^{2} - kx.ky)

F(kx,ky) = k^{4}/k^{2} . (2x^{2}y^{2})/(x^{2}+y^{2}) = k^{2 }(2x^{2}y^{2})/(x^{2}+y^{2}) â‡’ F(kx,ky) = k^{2}F(x,y)

G(kx,ky) = k^{2}/k^{2} . y^{2}/(x^{2} - xy) = k^{0}. y^{2}/(x^{2} - xy) â‡’ G(kx,ky) = k^{0}G(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) = x^{n}g(y/x)

F(x,y) = y^{n}h(x/y)

n = Degree of homogeneous equation

We consider the first function F(x,y) = (2x^{2}y^{2})/(x^{2}+y^{2})

= (2x^{2}[x^{2}.(y/x)^{2}])/(x^{2}.(1+ y^{2}/x^{2})) = (2x^{4}.(y/x)^{2}])/(x^{2}.(1+ y^{2}/x^{2}))

= (x^{2}. 2(y/x)^{2}])/(1+ (y/x)^{2}) = x^{2}F_{1}(y/x)

This is a homogeneous function of degree two.

Similarly, G(x,y) = y^{2}/(x^{2}-xy)

= y^{2}/(y^{2}[(x/y)^{2} - (x/y)]) = y^{2-2}. 1/[(x/y)^{2} - (x/y)] = y^{0} G_{1}(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 = x^{0}h(y/x) ................... (i)

Substitute y = vx.

dy/dx = v + x. dv/dx

equation (i) can be written as

v + x . dv/dx= x^{0}h(v) Or v + x dv/dx = h(v)

x dv/dx = h(v) - v

â‡’ 1/(h(v) - v) dv = 1/x dx

â‡’ âˆ« 1/(h(v) - v) dv = âˆ« 1/x dx

Replace v = y/x

For the homogeneous function in the differential equation is of the form dy/dx = y^{0}h(x/y)

Substitute x = vy in the equation.