Boundary Value Problem:

The given differential equation is y" xy = 0, where x belongs to [a,b] and y(a) = y(b) = 0. We need to determine the properties of the solution y to this boundary value problem.

1. Non-Trivial Solution:

A non-trivial solution means that y is not identically zero. In this case, if y is identically zero, then the equation y" xy = 0 is satisfied for all x in the given interval. However, the boundary conditions y(a) = y(b) = 0 cannot be satisfied by the zero function. Therefore, the solution y must be non-trivial.

2. Analysis of the Differential Equation:

The given differential equation is y" xy = 0. To analyze this equation, we can rewrite it as y" = 0 for x ≠ 0. This implies that the second derivative of y is zero, which means y is a linear function.

3. Implications of Boundary Conditions:

The boundary conditions y(a) = y(b) = 0 indicate that the solution y must cross the x-axis at both a and b. Since y is a linear function, it can be written as y = mx + c, where m and c are constants. From y(a) = 0, we have ma + c = 0, and from y(b) = 0, we have mb + c = 0. Subtracting these two equations, we get m(b-a) = 0, which implies m = 0 or b = a.

4. Implications of m = 0:

If m = 0, then y = c, where c is a constant. However, this solution does not satisfy the boundary conditions y(a) = y(b) = 0 unless c = 0. Therefore, the constant solution y = 0 is not a non-trivial solution.

5. Implications of b = a:

If b = a, then the interval [a, b] collapses to a single point, and the boundary value problem becomes y" xa = 0 with y(a) = y(a) = 0. In this case, the solution is y = 0, which is not a non-trivial solution.

Conclusion:

Based on the analysis, we can conclude the following:

a) The condition b > 0 is not directly related to the solution y of the given boundary value problem.

b) The solution y is not monotonic in the interval (a, b) because it is a linear function.

c) The derivative of y, y'(x), is a constant, and at x = a, we have y'(a) = m. Therefore, y'(a) = 0 only if m = 0, which is not a non-trivial solution.

d) The non-trivial solution y must satisfy the boundary conditions y(a) = y(b) = 0. Since y is a linear function, it can intersect the x-axis at most once. Therefore, y can have at most one zero in the interval [a, b].

In conclusion, the correct option is d) y has infinitely many zeros in [a, b] is false.