Show that the homogeneous equation (Ax^2+By^2)dx+(Cxy+Dy^2)d

  • #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.
 
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Answers and Replies

  • #2
36,063
7,997

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.
You have shown that if the equation is exact, then 2B = C.
It's easy to show that if 2B = C, then the equation is exact (i.e., that ##M_y = N_x##).
For this one, however, since each of the steps is really if and only if (reversible), you don't need to prove both statements.
 
  • #3
1,234
403
You have shown that if the equation is exact, then 2B = C.
It's easy to show that if 2B = C, then the equation is exact (i.e., that ##M_y = N_x##).
For this one, however, since each of the steps is really if and only if (reversible), you don't need to prove both statements.

Thanks for the clarification. So it is basic substitution of 2B=C in the equation to get ##M_y = N_x##). Thanks, Mark. I was overthinking the question.
 

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