Solve the given differential equation. \( 6 x d x+d y=0 \) \( y=\square \)

To solve the differential equation \( 6x \, dx + dy = 0 \), we can rearrange this into a more familiar form: \[ dy = -6x \, dx \] Now, we integrate both sides: \[ \int dy = \int -6x \, dx \] This gives us: \[ y = -3x^2 + C \] where \( C \) is the constant of integration. So, the solution to the differential equation is: \[ y = -3x^2 + C \] Now, let's explore some related aspects! The concept of differential equations dates back centuries, being a vital tool in physics, engineering, and economics. Early mathematicians like Newton and Leibniz laid the groundwork for calculus, which paved the way for these equations. Their work allowed us to model rates of change, like how a car accelerates or how populations grow, beautifully combining theory with real-world scenarios! In the realm of practical application, differential equations frequently pop up in physics. For instance, they can describe the motion of objects under the influence of forces, such as gravity or friction. Engineers use them to model everything from electrical circuits to heat transfer. So, the next time you notice a swinging pendulum or a flickering light bulb, remember that those phenomena are orchestrated by the elegant dance of differential equations!