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).

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).

cdquestions Admin · Accepted Answer

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 $.

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).

Correct Answer: 1

Solution and Explanation

Top Questions on Differential Equations

Questions Asked in IIT JAM MA exam

For a > 0, let ya(x) be the solution to the differential equation\(2\frac{d^2y}{dx^2}-\frac{dy}{dx}-y=0\)satisfying the conditionsy(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).

Correct Answer: 1

Solution and Explanation

Top Questions on Differential Equations

Questions Asked in IIT JAM MA exam

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).