Question

(x+y)zp+(x-y)zq=x^2+y^2 solve this using lagrangian multipliers method ​

Answer 1

Answer:

x(y2+z)p−y(x2+z)q=(x2−y2)z

where, p=∂z∂x and q=∂z∂y

My attempt:

I start with Lagrange's auxiliary equation

dxx(y2+z)=dy−y(x2+z)=dzz(x2−y2)

Relation 1:

xdx+ydy−dzx2y2+x2z−y2x2−y2z−x2z+y2z=xdx+ydy−dz0

Integrating xdx+ydy−dz=0

x22+y22−z=c⟹x2+y2−2z=c1

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