Exact differential equations With Relevent Questions

Exact Differential Equations:

            The differential equation of the form M(x,y)dx+ N(x,y)dy=0 is called exact differential equation if δM/δy=δN/δx.
        And the differential equations of the form M(x,y)dx+ N(x,y)dy=0 is called non exact differential equations if δM/δy≠δN/δx.
      And it's solution is given by
 ∫M(x,y)dx + ∫N(x,y)dy=c
    y as.           Terms don't contain
                                      x
constant
Let's discuss some examples related to this form,

Example :

     1. Solve the differential equation

    ( ycosx+ siny+y)dx+(sinx+ x cosy+x)dy=0 if it is exact.

Solution:

           Here M= ycosx+siny+y and N=sinx+xcosy+x
 Therefore δM/δy= cosx+cosy+1 and δN/δx=cosx+cosy+1
 Since δM/δy=δN/δx  the given equation is exact.
 Then it's solution is given by
      ∫Mdx+∫Ndy=c
=> ∫(y cosx+ siny+y)dx+∫0dy=c
=> y sinx+ x siny+xy+c1=c
=> y sinx+ x siny + xy= c
    Hence which is the required solution of the given differential equation.
I hope this may useful.

Try It By Yourself:

 (1) Test the equations

           eʸdx + (xeʸ + 2y)dy =0

 for exactness, and solve it if it is exact.

(2)   Solve 

           xe^(x²+y²) dx + y(e^(x²+y²) +1)dy =0

     with y(0) = 0.

(3) Solve the following problem.

   (cos x + y sin x) dx = cos x dy ,y(π) = 0.

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