Question

Identify the class (or method to use) for the equation and solve
#1) sin(x+y)dx + [2y+sin(x+y)]dy=0

#2) dy/dx=(sinx)y+2xe^(-cosx)

Thank you in advance for the help! Really stuck on these two!

Answer 1

1) might be an exact equation. Not sure.

Answer 2

I know the method and the solutions, I just don't know how to solve it..
#1 is exact and the general solution is y^2-cos(x+y)=constant

#2 is linear with the general solution y=[e^(-cosx)](x^2+C)

If anyone could show me the steps to get these answers, I would greatly appreciate it!! thank you!

Answer 3

#1) sin(x+y)dx + [2y+sin(x+y)]dy=0

compare wid below :
\(\large \mathbb{f_x dx + f_y dy = 0}\)

you need to find some function \(\large \mathbb{ f(x,y)}\), such that :
\(\large \mathbb{f_x = \sin(x+y)}\)
\(\large \mathbb{f_y = 2y+\sin(x+y)}\)