math unit 3 forumlas

Question 1

what is a homogeneous PDE and define the order and degree and when its linear

what are the standard notations used

Answer 1

linear: of its first degree in dependent var and its partial derivatives

homogeneous: can contain either variable or one of its partial derivative or both

order: the power to which highest order partial derivative is raised
ex: Uxx

order: highest order of derivatives:
Ux^2y

r=∂^2z/∂x^2
s=∂^2z/∂x∂y
t=∂^2z/∂y^2
p=∂z/∂x
q=∂z/∂y

2Z = x^2/a^2 + y^2/b^2
z∂z/∂x = 2x/a^2
z∂z/∂y = 2y/b^2
2 M ISA!!

Question 2

f=> (u,v) => (x,y,z)

Answer 2

for x
∂f/∂u[∂u/∂x∂x/∂x+∂u/∂y∂y/∂x+∂u/∂z*∂z/∂x]

∂f/∂v[∂v/∂x∂x/∂x+∂v/∂y∂y/∂x+∂v/∂z*∂z/∂x]

∂x/∂x = 0 and ∂y/dx =0

for y
∂f/∂u[∂u/∂x∂x/∂y+∂u/∂y∂y/∂y+∂u/∂z*∂z/∂y]=0

∂f/∂v[∂v/∂x∂x/∂y+∂v/∂y∂y/∂y+∂v/∂z*∂z/∂x]=0