Solve the given differential equation. \( 2 d y+\ln x y d x=(6 x+\ln y) d x \) \( y=\square \)

To solve the differential equation \( 2 dy + \ln(x)y \, dx = (6x + \ln(y)) \, dx \), we first rearrange it into a more standard form. This gives us: \[ 2dy = (6x + \ln(y) - \ln(x)y) \, dx \] Now, isolate \( \frac{dy}{dx} \): \[ \frac{dy}{dx} = \frac{(6x + \ln(y) - \ln(x)y)}{2} \] This is a nonlinear first-order differential equation and might require substitutions or special techniques. For real-world application, differential equations like this one model various phenomena, such as population dynamics, chemical reactions, and thermal processes. Engineers and scientists use these equations to predict future behavior in systems based on current states. When approaching differential equations, a common mistake is neglecting to check if the equation can be simplified or if any substitutions can be made to make the equation separable. Remember, careful manipulation can often reveal simpler underlying structures! Let's summarize the solution attempt: We rearranged the equation, prepared to isolate variables, and also noted the relevance of such equations in practical scenarios while highlighting the importance of precision in solving them!