Question

Solve the differential equation:
\[ \frac{d^2y}{dx^2} - 4y = 0 \]

• A. $y = C_1 e^{2x} + C_2 e^{-2x}$
• B. $y = C_1 \cos(2x) + C_2 \sin(2x)$
• C. $y = C_1 e^{x} + C_2 e^{-x}$
• D. $y = C_1 e^{4x} + C_2 e^{-4x}$

Hint:

For second-order ODEs, the form of the solution depends on the roots of the characteristic equation: real distinct, real equal, or complex.

Answer 1

The Correct Option is A

This is a second-order linear homogeneous differential equation.
The characteristic equation is:
\[ r^2 - 4 = 0 \]
\[ r = \pm 2 \]
Since the roots are real and distinct, the general solution is:
\[ y = C_1 e^{2x} + C_2 e^{-2x} \]
Verify: $y' = 2C_1 e^{2x} - 2C_2 e^{-2x}$, $y" = 4C_1 e^{2x} + 4C_2 e^{-2x}$.
\[ y" - 4y = (4C_1 e^{2x} + 4C_2 e^{-2x}) - 4(C_1 e^{2x} + C_2 e^{-2x}) = 0 \]
Option (1) is correct.