Question

ques

Answer 1

How can I solve the following?:
\[xdx+ydy+\frac{xdy-ydx}{x^2+y^2}=0\]
I multiplied with x^2+y^2 throughout
\[(x^3+xy^2-y)dx+(x^2y+y^3+x)dy=0\]
Now we have
\[\frac{\partial M}{\partial y}=2xy-1\]\[\frac{\partial N}{\partial x}=2xy+1\]
None of the factors
\[\frac{\frac{\partial M}{\partial y}-\frac{\partial N}{\partial x}}{N} \space \space \space , \space \space \space \frac{\frac{\partial N}{\partial x}-\frac{\partial M}{\partial y}}{M}\]
seem to simplify the problem and it doesn't look like homogenous so
\[\frac{1}{Mx+Ny}\]
seems to be out of question

Answer 2

@Nishant_Garg

Answer 3

yes?

Answer 4

Ok, but you can't always just recognize the differentials, isn't there some method for this though?

Answer 5

How would I know that I can write xdx+ydx exactly as 1/2 the differential of x^2+y^2...

Answer 6

check any standard book of differential equations , it has shortcut formulae, nearly 20
and see d(x^2)=2xdx so (1/2)d(x^2)=xdx similarly (1/2)d(y^2)=ydy
add these two and you will get (1/2)d(x^2+y^2)=xdx+ydy

Answer 7

If you have any question let me know..
And those shortcut equivalent differentials you will have to commit to memory as it will save tons of time in the future