Class 12 Maths Differential Equations Homogeneous Differential Equation

Homogeneous Differential Equation

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

Example: Below Equation is Homogeneous Differential Equation

          

Since,

F(λx, λy) = λ0F(x,y)  

i.e  F(λx, λy) = F(x,y)

 

Method to solve a given homogeneous differential equation

     ………(1)

We make the substitution of    y = v . x  ……….(2)

Differentiating equation (2) with respect to x, we get

      ……..(3)

 

Substituting the value of  dy/dx  from equation (3) in equation (1), we get

   ………(4)

 

Arranging the variables,                                          ………(5)

 

Equation (5) gives general solution (primitive) of the differential equation (1) when we replace v by y/x

Note: If the homogeneous differential equation is in the form dy/dx= F(x, y) where, F (x, y) is homogenous function of degree zero, then we make substitution of x/y = v   i.e., x = vy and we proceed further to find the general solution as discussed above by writing dy/dx = (x,y) = g(y/x)

 

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