I am trying to understand example 1.7 on "Differential Equations Desmistified" book but I can't understand how the author convert (x/2)*cot(y)dy/dx=-1 into 2X*sin(y)dx+x^2*cos(y)dy=0..Can you please explain how? thank you very much

Question

mathstudent55 · Answer

cot(y) = cos(y)/sin(y)

dumbcow · Answer

$$\frac{x \cos(y) dy}{2 \sin(y) dx} + 1 = 0$$
combine into single fraction
$$\frac{x \cos(y) dy + 2 \sin(y) dx}{2 \sin(y) dx} = 0$$
...

dumbcow · Answer

oh and then multiply equation by "x"

mathstudent55 · Answer

$\dfrac{x}{2} \cot y \dfrac{dy}{dx}=-1$

$\dfrac{x}{2} \dfrac{\cos y}{\sin y} \dfrac{dy}{dx}=-1$

Multiply both sides by $2 \sin y ~dx$

$x \cos y dy = -2 \sin y ~dx$

Add $2 \sin y ~dx$ to both sides.

$2 \sin y ~dx + x \cos y ~dy = 0$

Multiply both sides by x:

$2x \sin y ~dx + x^2 \cos y ~dy = 0$

anonymous · Answer

Thank you very much!!! to all of you! I just discovered this site and I love it.

mathstudent55 · Answer

You are welcome.