Question

(xz)dx+(zy)dy=(x^2+y^2)dz

Answer 1

Given differential equation is

$xzdx+zydy=(x^2+y^2)dz$

$z(xdx+ydy)=(x^2+y^2)dz$

$\frac{xdx+ydy}{x^2+y^2}=\frac{dz}{z}$ .....(1)

let $x^2+y^2=t$

2xdx+2ydy=dt

Substitute in equation 1

$\frac{1}{2}\frac{dt}{t}=\frac{dz}{z}$

Integrating Both the side

$\frac{1}{2}\int\frac{dt}{t}=\int\frac{dz}{z}$

$\frac{1}{2}logt=logz+logc$

logt$^\frac{1}{2}=logzc$

$t^\frac{1}{2}=zc$

$(x^2+y^2)^\frac{1}{2}=zc$