- #1

- 1,234

- 403

## Homework Statement

Show that the homogeneous equation: $$(Ax^2+By^2)dx+(Cxy+Dy^2)dy=0$$ is exact iff 2b=c.

## Homework Equations

None, just definitions.

## The Attempt at a Solution

Let $$M = Ax^2+By^2$$ and $$N = Cxy+Dy^2$$

Taking the partial derivative of M with respect to y and the partial of N with respect to x we get

$$\frac{\partial M}{\partial y} \ =2By$$ and $$\frac{\partial N}{\partial x} \ =Cy$$

$$\frac{\partial M}{\partial y} \ =\frac{\partial N}{\partial x} $$ is true only if 2B=C.

What is giving problems here is the iff statement. Can I compete this problem by stating that 2b=c, then

$$\frac{\partial M}{\partial y} \ =\frac{\partial N}{\partial x} $$ ??

Can someone point me in the right direction.

Last edited by a moderator: