Question

Solve patial diffrential eq
(y^2)p - xyq = x( z-2y )

Answer 1

Patial diffrential eq uation of : y²p-xyq = x(z-2y)

Step-by-step explanation:

Given y²p-xyq = x(z-2y)

The form of the equation is Pp+Qq = R

» P = y² ,

»Q = -xy ,  

» R = x(z-2y)

The Lagranges’s Subsidiaryu equation are dx/P = dy/Q = dz/R

i.e, dx/y² = dy/-xy = dz/x(z-2y)

STEP 1:

dx/y² =dy/-xy

dx/y = dy/-x

xdx = -ydy

ʃ x dx = - ʃy dy

x²/2 = - y²/2 + c1/2

x² + y² = c1

u= x² + y²

STEP 2 :

dy/-xy = dz/x(z- 2y)

dy/-y = dz/(z-2y)

(z-2y)dy = -ydz

z dy – 2y dy + - y dz

Ydz + zdy = 2ydy

ʃd (yz) = ʃ 2y dy

yz = y² + c2

v = yz-y²

Therefore the solution for the given equation is f(x² + y², yz-y²) = 0