Question

For the ordinary differential equation \( \frac{d^2y}{dx^2} + 4y = 0 \), the general solution is:

• A. \( y = c_1 \cos 2x + c_2 \sin 2x \)
• B. \( y = c_1 \cosh 2x + c_2 \sinh 2x \)
• C. \( y = c_1 e^{2x} + c_2 e^{-2x} \)
• D. \( y = c_1 e^{2x} \cos 2x + c_2 e^{-2x} \sin 2x \)

Hint:

For second-order equations like \( y'' + a^2 y = 0 \), if roots are \( \pm bi \), always write the solution as \( y = c_1 \cos bx + c_2 \sin bx \).

Answer 1

The Correct Option is A

Step 1: Start with the given second-order linear homogeneous differential equation:
\[ \frac{d^2y}{dx^2} + 4y = 0 \]
Step 2: Form the characteristic (auxiliary) equation:
\[ r^2 + 4 = 0 \]
Step 3: Solve the quadratic equation:
\[ r = \pm \sqrt{-4} = \pm 2i \]
These are purely imaginary roots.

Step 4: The general solution for roots of the form \( \pm bi \) is:
\[ y(x) = c_1 \cos bx + c_2 \sin bx \]
Here, \( b = 2 \), so:
\[ y(x) = c_1 \cos 2x + c_2 \sin 2x \]