 #1
 95
 9
Homework Statement:

$$yx'_x  xz'_y = xyz$$
Solve by variable change:
$$u = x^2 + y^2$$
$$v = e ^{x^2/2}$$
Relevant Equations:
 None.
I completely forgot how to solve these so heres my attempt:
$$z = au + bv$$
$$z = a(x^2 + y^2) + be ^{x^2/2}$$
$$z'_x = 2ax  bxe ^{x^2/2}$$
$$z'_y = 2ay$$
Put that into the original equation and you get
$$y * (2ax  bxe ^{x^2/2}) x * (2ay) = $$
$$ybe^{x^2/2} = xyz$$
$$z = be^{x^2/2}/x$$
But the solution should be
$$e^{x^2/2} * f(x^2 + y^2)$$
Did I do everything wrong or just missed something here?
$$z = au + bv$$
$$z = a(x^2 + y^2) + be ^{x^2/2}$$
$$z'_x = 2ax  bxe ^{x^2/2}$$
$$z'_y = 2ay$$
Put that into the original equation and you get
$$y * (2ax  bxe ^{x^2/2}) x * (2ay) = $$
$$ybe^{x^2/2} = xyz$$
$$z = be^{x^2/2}/x$$
But the solution should be
$$e^{x^2/2} * f(x^2 + y^2)$$
Did I do everything wrong or just missed something here?