Final answer:
The problem is to find the general solution of the differential equation e^x dy + (y e^x + 2x) dx = 0, which can be solved by checking exactness or finding an integrating factor.
Step-by-step explanation:
To find the general solution of the given differential equation e^xdy + (ye^x + 2x)dx = 0, we can rearrange it and integrate.
e^xdy + (ye^x + 2x)dx = 0
e^xdy = - (ye^x + 2x)dx
Now, integrate both sides:
int e^y ,dy = - \int (ye^x + 2x) ,dx
Integrate the left side with respect to \(y\):
e^y = - int (ye^x + 2x) ,dx + C
Now, integrate the right side with respect to x:
e^y = - int ye^x ,dx - int 2x ,dx + C
e^y = -e^x(y + 2x) + C
Solve for y:
y = -2x - 1 + Ce^{-x}
So, the general solution to the given differential equation is:
y = -2x - 1 + Ce^{-x}
where C is an arbitrary constant.