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 o