Mathquery

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+cos...

Homogeneous Differential Equation:             Now I want to different forms and types of differential equations .First of all we should focus on the types, that are Homogeneous Differential equations, Exact and non exact differential equations etc. The homogeneous equation must have same degree.   For example :  solve                x^2y dx - ( x^3+y^3) dy = 0 Solution :              The given equation can be written in the form          dy/ dx = x^2y/x^3+y^3 Putting y= vx, we have   v+xdv/dx= x^2.vx/x^3+v^3x^3 = v/ 1+v^3 => x dv/dx = v/1+v^3 - v                     = -v^4/1+v^3 =>(1+v ³/v⁴)dv = - dx/x =>(1/v⁴+1/v)dv = - dx/x Integrating,      -1/3v³+log(v) = - log(x)+log(c) => log(vx/c)=1/3v³ => vx/c = exp(1/3v³) =...

Solution Of Differential Equations :     From ancient Greek Mathematics equations are the short form of a long analytical problem .  A large problem can be solved in a easy way by converting it into a simple equation.  Differential equations are the evolved form of equations .   So here i want to discuss about some problems of Differential equations and it's methods of solving problems .         First of all I want to say something about its definition.Its definition is that it is the equation containing or having differential coefficient .     For example_ :                   d^2y/dx^2+dy/dx+y=3     Let us consider some problems and solutions regarding this.   Examples :          1. d^2y/dx^2+y=x          2. y"+y'+6y=0 Solutions :           1. Given ...