Homogeneous Differential Equations
A differential equation of the form dy/dx = f (x, y)/ g (x, y) is called homogeneous differential equation if f (x, y) and g(x, y) are homogeneous functions of the same degree in x and y. (or)
Homogeneous differential can be written as dy/dx = F(y/x).
Check f ( x, y) and g ( x, y) are homogeneous functions of same degree.
The given differential equation becomes v x dv/dx =F(v)
Separating the variables, we get
By integrating we get the solution in terms of v and x.
Replacing v by y/x we get the solution.
Solve the differential equation y2 dx + ( xy + x2 )dy = 0
Find the particular solution of the differential equation x 2 dy + y ( x + y) dx = 0 given that x = 1, y = 1
If the marginal cost of producing x shoes is given by (3xy + y2 ) dx + (x 2 + xy) dy = 0 and the total cost of producing a pair of shoes is given by ₹12. Then find the total cost function.
Given marginal cost function is (x2 + xy) dy + (3xy + y2)dx=0
The marginal revenue ‘y’ of output ‘q’ is given by the equation . Find the total Revenue function when output is 1 unit and Revenue is ₹5.