What is the solution to the following differential equation, subject to the initial condition \( y(0)=2 \) ? \( \frac{d y}{d x}=6 x+6 \)

To solve the differential equation \( \frac{d y}{d x} = 6 x + 6 \), we can integrate both sides with respect to \( x \). The integral of \( 6x \) is \( 3x^2 \), and the integral of \( 6 \) is \( 6x \). Therefore, \[ y = 3x^2 + 6x + C \] Next, we can apply the initial condition \( y(0) = 2 \). Substituting \( x = 0 \): \[ y(0) = 3(0)^2 + 6(0) + C = 2 \implies C = 2 \] Thus, the solution to the differential equation, with the initial condition, is \[ y = 3x^2 + 6x + 2. \] The above function is a quadratic equation, and it represents a parabola that opens upwards. It varies with \( x \), demonstrating how the output value \( y \) changes corresponding to different inputs of \( x \). This can be useful in various real-world scenarios, such as physics for motion problems where position is described in terms of time. It's also important to remember that while integrating, you might have a tendency to forget the constant \( C \). Be sure to always apply your initial conditions correctly after finding the general solution to avoid errors. Common mistakes often arise from neglecting \( C \) or miscalculating it when substituting values!