Question

Find the largest interval which includes x = 0 for which the given initial-value problem has a unique solution. (Enter your answer using interval notation.) (x − 4)y'' + 5y = x, y(0) = 0, y'(0) = 1

Answer 1

The largest interval that includes x = 0 for which the given initial-value problem has a unique solution is (4, ∞).

Step-by-step explanation:

To find the largest interval which includes x = 0 for which the given initial-value problem has a unique solution, we can use the existence and uniqueness theorem for second-order linear homogeneous differential equations. The given equation is (x − 4)y'' + 5y = x, with initial conditions y(0) = 0 and y'(0) = 1.

To apply the theorem, we first divide the equation by (x - 4) to get y'' + (5/(x - 4))y = x/(x - 4). Now, let's analyze the interval of (x - 4). When x - 4 > 0, the interval is (4, ∞). When x - 4 < 0, the interval is (-∞, 4).

Since we are interested in finding the interval that includes x = 0, we need the interval where x - 4 > 0, which is (4, ∞). Therefore, the largest interval that includes x = 0 for which the given initial-value problem has a unique solution is (4, ∞).

Answer 2

the answer is 46^  y equals o