Question

Please help me solve this problem. Transform the equation to new variables (u and v). How do you do this?

`2x (∂z)/(∂x) - y (∂z)/(∂y)=0`
when
`x= u^2v, and y=1/u`

Answer 1

For `z = z(x,y)`:

• `2x \ bbz_x - y \ bbz_y=0 qquad square`

The substitutions:

`{(x= u^2v),(y=1/u):} qquad harr qquad {(u = 1/y),(v = xy^2):} to {(u_x = 0, u_ y = - 1/y^2),(v_x = y^2, v_y = 2xy):}`

The partials:

• `bbz_x = z_u u_x + z_v v_x`

`= z_v y^2 = z_v/u^2`

• `bbz_y = z_u u_y + z_v v_y`

`= - 1/y^2 z_u + 2xy z_v`

`= - u^2 z_u + 2uv z_v`

Plugging in:

`square implies 2 u^2 v z_v/u^2 - 1/u( - u^2 z_u + 2uv z_v ) = 0`

`z_v cancel (( 2 v - 2v)) + u z_u = 0`

`implies z(u,v) = z(v) = z(xy^2)`

Any function with the argument `xy^2` should satisfy this PDE