Solve differential equation : x(3x+2y^2)dx+2y(1+x^2)dy=0

Question

This is to be done with linear equations form. 

Answer 1

M(x,y) = 3x  + 2x*y^2 

 

dM/dy = 4xy

 

 

N(x,y) = 2y + 2x^2*y

dN/dx = 4xy

 

Since the partial derivative of M w.r.t y is the same as the partial derivative of N w.r.t x

the differential equation is exact

 

integrating M w.r.t x:

  integral (  3x + 2x*y^2 ) dx  =  (3/2)x^2 + x^2*y^2 + C(y)  where C(y) is a constant function of y;

 This is the particular solution F(x,y) 

 

So,

dF/dy = 2x^2*y + C'(y) =  2x^2*y + 2y = N(x,y)

 

    Then C'(y) = 2y

     C(y) = y^2

 

The implicit solution is then  (3/2)x^2 + x^2*y^2 + y^2 + k