Answer to Question #94680 in Differential Equations for Deepak

Question

solve differential (y^2+yz)dx+(xz+z^2)dy+(y^2-xy)dz=0 please sir tell name of the form of which differential equation.and explain

Answer 1

Solution of Pfaffian Differential equation in three variables.

Verify the Pfaffian Differential equation

$$(y^2+yz)dx+(xz+z^2)dy+(y^2-xy)dz=0$$

is integrable and find its prmitive.

The necessary and sufficient condition for iintegrability is

$$\bm{X}\cdot curl \bm{X}=0$$

$\bm{X}=(y^2+yz,xz+z^2,y^2-xy)$ so that

$$\bm{ \nabla}\times\bm{X}=\begin{vmatrix} \bf{i} & \bf{j} & \bf{k} \\ {\partial\over \partial x} & {\partial\over \partial y} & {\partial\over \partial z} \\ y^2+yz & xz+z^2 & y^2-xy \end{vmatrix}=$$ $$=(2y-x-x-2z){\bf{i}}+(y+y){\bf{j}} + (z-2y-z){\bf{k}}=$$ $$=(2y-2x-2z){\bf{i}}+(2y){\bf{j}} + (-2y){\bf{k}}$$

$$\bm{X}\cdot (\bm{ \nabla}\times\bm{X})=2y^3-2xy^2-2y^2z+2y^2z-$$ $$-2xyz-2yz^2+2xyz+2yz^2-2y^3+2xy^2=0$$

Thus the given equation is integrable.

Solve by Inspection

$$y(y+z)dx+z(x+z)dy+y(y-x)dz=0$$

Or

$$y(y+z)dx+y(y+z)dz-y(y+z)dz+$$ $$+z(x+z)dy+y(x+z)dy-y(x+z)dy+$$ $$+y(y-x)dz=0$$

Or

$$y(y+z)d(x+z)+(y+z)(x+z)dy-$$ $$-ydz(y+z-y+x)-y(x+z)dy=0$$

Or

$$y(y+z)d(x+z)+(y+z)(x+z)dy-y(x+z)d(y+z)=0$$

$${d(x+z) \over x+z}+{dy\over y}-{d(y+z) \over y+z}=0$$

The complete primitive is given as

$$y(x+z)=c(y+z)$$