Word Count : 581

---

To solve the given differential equation \((6xy - 3y^2 + y)dx + 2(x - y)dy = 0\), we will first find its integrating factor and then solve it. ### Step 1: Check for exactness The given differential equation is: \[ (6xy - 3y^2 + y)dx + 2(x - y)dy = 0 \] We can identify the components as: \[ M(x, y) = 6xy - 3y^2 + y \quad \text{and} \quad N(x, y) = 2(x - y) \] To check if the equation is exact, we need to compute the partial derivatives of \(M\) and \(N\) with respect to \(y\) and \(x\) respectively. - \(\frac{\partial M}{\partial y} = \frac{\partial}{\partial y}(6xy - 3y^2 + y) = 6x - ___ ___ ______ _____ _________ ________ ____ _______ _____ ______ ________ _________.
___ _________ ___ ______ _________ __________ __________ ______ ____ ___ ___.
_________ ____ _____ ____ __________.
_____ __________ _____ ________ _______ _______ _________.
_______ ______ __________ ________ ___ _______ ______ _____ ___ ________ ______ __________.
____ ______ ________ __________ ______.
__________ _____ __________ _______ _________ ________ __________ ________.
________ _________ _________ ___ _______ ___ ______ _____ __________.
_____ _______ _____ ___ _______ _________ __________ ______ _____ _______ _________ ___.
___ __________ ___ ________ _________ _________ __________ _____ __________ __________.
________ ______ ________ ________ ___ ____ ______ ______.
___ _______ _____ _____ __________.
___ ___ ____ _________ ________.
_______ ________ ______ ____ ________ _____.
__________ _____ _____ _________ _______.
_______ ________ ________ _____ _________ _________ ______ ___.
_______ ________ ______ _____ __________ _________ ________ _________ ____ ___ __________ ____.
_____ _______ _______ __________ ________ ________ _________ ___ ___ ___ ________.
_______ ________ ________ _______ ________ ____ ___ _________ __________ ________.
_________ ______ ____ ___ ____ _________.
__________ _____ _______ ______ ______ _________ ____ ___ _______ _____ _________.
_______ ____ ____ ______ _______ _________ ________ ________ ____ ___.
_____ ___ _______ __________ _______ _______ ________ __________.
________ ________ ______ ________ ___ __________ _________ __________ ___ _______.
________ _________ _________ ______ _____ ______.
_________ _____ _________ ____ __________ __________ ___ _________ ________ _______.
______ ________ __________ _______ ___ ____ _____ _______ ______.
______ _______ ________ ____ _____ ______ _________ ____ ___ ____.
____ ____ _____ ________ ________.
_______ __________ __________ _________ __________ __________ ______ ________ ______ ______ ____.
__________ ______ _______ __________ _______.
_______ ___ __________ ________ _______ _____ _________ __________ ___ _____ ____.
___ _______ _________ ______ _______ ________ ________ _____ ________.
____ _______ ________ ________ ________ ____ ______ ____ ___ ___ __________ ______.
________ _______ _________ ____ ______ ___ ____ _____ _______ ______.
____ __________ _____ ____ ______.
_____ _______ ___ ___ __________ ________ _________ ____ ______ _____ _________.
__________ _____ ________ _______ _____ ________ _____ ____ ____.
______ _________ ______ ________ _______ _________ ____ _____.
__________ ____ ___ ______ _____ _______ __________ __________ _________ ________ ______.
___ _____ ____ _________ _______ __________ ____ ________.
________ __________ ______ ______ __________ _____ ____ __________ __________ ___.
____ ______ _______ ______ ____ _____ __________ ____.
___ ______ __________ ________ _______ _________ _______.
__________ _________ ___ _____ ________.
__________ ________ _______ ___ __________ _______ ______ _____.
________ ___ ____ ________ ___ ________ ________ __________ ________ _________ ________ ____.
_____ __________ _________ ________ ___ _____ ____ ________ ___ ______.
_____ _________ ______ _______ _________ ______ _____ _________ _____ ____.
_____ _____ _________ ____ __________.
___ ____ ____ ____ ___ ______ ___ __________ _____ ________ ___.
____ ___ ______ _________ ______.
_____ _________ _____ ___ ____ ________ _____ _____ ______ _____ ______ __________.
____ ____ _________ ______ ______.
______.
Get Full Answer on WhatsApp