Question

For a > 0, let y_a(x) be the solution to the differential equation
\(2\frac{d^2y}{dx^2}-\frac{dy}{dx}-y=0\)
satisfying the conditions
y(0) = 1, y'(0) = a.
Then, the smallest value of a for which ya(x) has no critical points in ℝ equals ___________ (rounded off to the nearest integer).

Answer 1

Correct Answer: 1

To solve this second-order differential equation, we find the general solution and analyze the critical points by setting \( y'(x) = 0 \). The smallest value of \( \alpha \) that prevents any critical points is obtained through careful calculation, which gives \( \alpha = 1.0 \).