### Homogeneous Equations

**We have to understand about Homogeneous Equation in Calculus .**

If

**M**and

**N**of the equation

**Mdx + Ndy = 0**are both of the same degree in

**x**and

**y**, and are homogeneous , the equation is said to be homogeneous . Such an equation can be put in the form

**dy/dx = f ( y/x )**

Every homogeneous equation of the above type can be easily solved by putting

**y = vx**

where

**v**is a function of

**x**, and consequently

**dy/dx = v + x ( dv/dx )**

whereby it reduced to the form

**v + xdv/dx = f ( v )**

i.e.

**dx/x = dv/ [ f(v) - v ]**

in which the variables are separated

**A Special Form :-**

The equation of the form

**dy/dx = [ ( a1x +b1y +c1 ) / ( a2x + b2y +c2 ) ]**

**where , [ a1/a2 is not equal to b1/b2 ] --------------(1)**

can be easily solved by putting

**x = x' + h**

and

**y = y' + k**

where

**h**and

**k**are constant

So that ,

**dx = dx' and dy = dy'**

and choosing h and k in such a way that

**a1h + b1k + c1 = 0 and a2h + b2k + c2 =0 -------(2)**

For now the equation reduces to the form

**dy'/dx' = ( a1x' +b1y' ) / ( a2x' + b2y' )**

which is homogeneous in

**x'**and

**y'**and hence solved .

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