Question

Consider the differential equation \(\frac{d^2y}{dx^2} - 2 \frac{dy}{dx} + y = 0\). The boundary conditions are \(y = 0\) and \(\frac{dy}{dx} = 1\) at \(x = 0\). Then the value of \(y\) at \(x = \frac{1}{2}\) is

• A. 0
• B. \(\sqrt{e}\)
• C. \(\frac{\sqrt{e}}{2}\)
• D. \(\frac{e}{\sqrt{2}}\)

Hint:

For second-order linear differential equations with constant coefficients, solve the characteristic equation and apply the given boundary conditions to find the constants.

Answer 1

The Correct Option is C

Step 1: Solve the differential equation.
The given differential equation is: \[ \frac{d^2y}{dx^2} - 2 \frac{dy}{dx} + y = 0 \] The characteristic equation is: \[ r^2 - 2r + 1 = 0 \] Solving this gives \(r = 1\), so the solution to the differential equation is: \[ y = (C_1 + C_2x)e^x \] Step 2: Apply the boundary conditions.
Using \(y = 0\) at \(x = 0\), we find \(C_1 = 0\). Using \(\frac{dy}{dx} = 1\) at \(x = 0\), we find \(C_2 = 1\). So the solution is: \[ y = xe^x \] Step 3: Find \(y\) at \(x = \frac{1}{2}\).
Substitute \(x = \frac{1}{2}\) into the solution: \[ y\left( \frac{1}{2} \right) = \frac{1}{2}e^{1/2} = \frac{\sqrt{e}}{2} \]