Question

The solution of the differential equation \(\frac{d^2y}{dx^2} - 3 \frac{dy}{dx} + 2y = 0\) satisfying \(y(0) = 0\), \(y'(0) = 1\), is

• A. \( y(x) = e^x - e^{2x} \)
• B. \( y(x) = e^x + e^{2x} \)
• C. \( y(x) = -e^x - e^{2x} \)
• D. \( y(x) = -e^x + e^{2x} \)

Hint:

For second-order linear differential equations with constant coefficients, use the characteristic equation to find the roots, then apply initial conditions to find the particular solution.

Answer 1

The Correct Option is D

To solve this second-order differential equation, we first solve the characteristic equation: \[ r^2 - 3r + 2 = 0 \] Factoring gives us the roots \( r_1 = 1 \) and \( r_2 = 2 \). Thus, the general solution is: \[ y(x) = c_1 e^x + c_2 e^{2x} \] Using the initial conditions \( y(0) = 0 \) and \( y'(0) = 1 \), we solve for \(c_1\) and \(c_2\), giving us the solution: \[ y(x) = -e^x + e^{2x} \]