If $ y = y(x) $ is the solution of the differential equation, \[ \sqrt{4 - x^2} \frac{dy}{dx} = \left( \left( \sin^{-1} \left( \frac{x}{2} \right) \right)^2 - y \right) \sin^{-1} \left( \frac{x}{2} \right), \] where $ -2 \leq x \leq 2 $, and $ y(2) = \frac{\pi^2 - 8}{4} $, then $ y^2(0) $ is equal to:

Question

Given that the solution of \[ \frac{d^2y}{dx^2} + \alpha \frac{dy}{dx} + \beta y = -e^{-x} \] is \[ y(x) = C_1 e^{-x} + C_2 e^{2x} + x e^{-x}, \] find the values of $\alpha$ and $\beta$.

Answer 1

Step 1: Differential Equation Provided

The differential equation to be solved is: \[ \sqrt{4 - x^2} \frac{dy}{dx} = \left( \left( \sin^{-1} \left( \frac{x}{2} \right) \right)^2 - y \right) \sin^{-1} \left( \frac{x}{2} \right) \]

Step 2: Equation Rearrangement and Integration

The equation is rearranged, and then integrated to determine the solution for $ y(x) $: \[ y = \left( \sin^{-1} \left( \frac{x}{2} \right) \right)^2 - 2 + c \cdot e \]

Step 3: Constant Calculation Using Initial Condition

Using the initial condition $ y(2) = \frac{\pi^2}{4} - 2 $, the constant $ c $ is determined: \[ y(2) = \frac{\pi^2}{4} - 2 \implies c = 0 \]

Step 4: Determination of $ y(0) $

Consequently, the value of $ y(0) $ is: \[ y(0) = -2 \]

Final Result: $ y(0) = -2 $

Show Hint

Correct Answer: 2

Solution and Explanation

Top Questions on Differential Equations

Questions Asked in JEE Main exam