# Thread: Are these differential equations exact?

1. ## Are these differential equations exact?

Can someone show me how to do this step by step at least for one of them. please!!!

find if these diff equ are exact and if they are solve them

2. (2x +y)dx - (x+6y)dy = 0

3. (1+ln x+ (y/x))dx = (1-ln x)dy

any help is really appreciated
thank you

2. 2.

$\displaystyle (2x+y)dx - (x+6y)dy = 0$

$\displaystyle y' = \frac{2x+y}{x+6y}$

Let $\displaystyle y = u \cdot x$, then $\displaystyle y' = u'x+u$

$\displaystyle u'x+u = \frac{2x+ux}{x+6ux}$

$\displaystyle u'x+u = \frac{2+u}{1+6u}$

$\displaystyle u'x = \frac{2-6u^2}{1+6u}$

$\displaystyle \frac{du}{dx} x = \frac{2-6u^2}{1+6u}$

It's separable now.

3. Originally Posted by kithy
Can someone show me how to do this step by step at least for one of them. please!!!

find if these diff equ are exact and if they are solve them

2. (2x +y)dx - (x+6y)dy = 0

3. (1+ln x+ (y/x))dx = (1-ln x)dy

any help is really appreciated
thank you
DEs are exact when $\displaystyle \frac{\partial M}{\partial y}=\frac{\partial N}{\partial x}$

From #2, we see that $\displaystyle M(x,y)=2x+y\implies \frac{\partial M}{\partial y}=1$.

We also see that $\displaystyle N(x,y)=-x-6y\implies \frac{\partial N}{\partial x}=-1$

They are not exact since $\displaystyle 1\neq -1$.

You can go about solving it the way that wingless did.

However, #3 is exact. I'll let you show that on your own. (You can only perform the test for exact equations iff the DE is in the form $\displaystyle M(x,y)\,dx+N(x,y)\,dy=0$)

Hope this helps you out!

--Chris