Final answer:
Upon substituting and differentiating each of the given functions, none satisfies the differential equation y" + y = 7 sin(x), therefore, none of them are correct solutions.
Step-by-step explanation:
To determine if the given functions are solutions to the differential equation y" + y = 7 sin(x), we need to plug each function into the equation and check if it satisfies the equation.
- y = 7 sin(x): By differentiating twice, y" = -7 sin(x). Adding y and y" gives 0, not equal to 7 sin(x), so it is not a solution.
- y = 7 cos(x): Differentiating twice, y" = -7 cos(x), and y + y" = 7 cos(x) - 7 cos(x) = 0, which also does not satisfy the equation. Thus, it is not a solution.
- y = 7x sin(x) - 8x cos(x): This function is more complex. However, when we differentiate it twice and then add y, the result does not equal the right-hand side of the differential equation. Therefore, this function is not a solution.
- y = 7/2 sin(x): Again, differentiating twice, we get y" = -7/2 sin(x), and y + y" = 0 not equal to 7 sin(x). So, this function is not a solution either.
- Y = - 7/2 cos(x): Similarly, differentiating twice yields y" = -7/2 cos(x), and adding y and y" does not satisfy the original equation. Thus, it is not a solution.
None of the provided functions are solutions to the given differential equation.