Q. 174.0( 2 Votes )

Show that the differential equation representing one parameter family of curves (x2 – y2) = c(x2 + y2)2 is (x3 – 3xy2)dx = (y3 – 3x2y)dy.

Answer :

Given, A differential equation (x3 – 3xy2)dx = (y3 – 3x2y)dy

To Prove: Prove that (x2 – y2) = c(x2 + y2)2


Explanation: We have (x3 – 3xy2)dx = (y3 – 3x2y)dy


Now, Find dy/dx from the given equation



Let


Put x = λx and y = λy in F(x, y)



Taking λ3 as a common from numerator and denominator then,


F(x, y) = λ0F(x, y)


Since, F(x, y) is a homogenous function of degree zero.


Let us Assume , y = vx - - - (ii)


Differentiate equation (i) w.r.t x


- - - (iii)


Now, Comparing the equation (i) and (iii), we get



Taking x3 as common then, we get







Now, Integrating both sides




Let I = , then


I = log|x| + C


I =



Let 1 - v4 = t


Differentiating , - 4v3 dv = dt


And,



Now, put u = v2


On differentiating w.r.t v






We know,



Putting the values of t









Now, Putting v = y/x









Hence, Proved


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