Final answer:
After integrating the given differential equation and applying the initial condition (0,12), the correct solution is y = -cosx - x⁵/5 - 3e(-4x) + 16, which is not listed in the provided answer choices.
Step-by-step explanation:
The question asks to solve the differential equation y' = sinx - 5x⁴ + 12e(-4x) with the initial condition that the solution passes through the point (0,12). To integrate the given differential equation, we need to integrate the right-hand side with respect to x, which will give us the general solution for y. The antiderivative of sinx is -cosx, the antiderivative of -5x⁴ is -x⁵/5, and the antiderivative of 12e(-4x) is -3e(-4x). Therefore, the general solution is y = -cosx - x⁵/5 - 3e(-4x) + C, where C is the constant of integration.
Using the initial condition that when x = 0, y = 12, we substitute these values into the general solution to solve for C: 12 = -cos(0) - 0 - 3e(0) + C, which simplifies to 12 = -1 - 3 + C, hence C = 16. The specific solution that satisfies the given initial condition is y = -cosx - x⁵/5 - 3e(-4x) + 16. None of the provided answer choices match this solution. To solve the differential equation y' = sinx - 5x⁴ + 12e(-4x) and find the solution that passes through the point (0,12), we can use the method of integration. Here are the steps: Integrate both sides of the equation with respect to x. The integral of y' with respect to x is simply y. Integrate sinx, -5x⁴, and 12e(-4x) separately. Apply the initial condition by substituting x = 0 and y = 12 into the equation. Solve for the constant of integration. Combine the results to get the final solution. The correct solution to the differential equation that passes through the point (0,12) is y = sinx - 5x⁴ + 12e(-4x).