- #1
Addez123
- 199
- 21
- 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 here's 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?