Question

If \( c \) and \( d \) are arbitrary constants, then \[ y = e^{2x} \left( \cosh \sqrt{2} x + d \sinh \sqrt{2} x \right) \] is the general solution of the differential equation:

• A. \( y'' + 4y' + 2y = 0 \)
• B. \( y'' - 4y' + 2y = 0 \)
• C. \( y'' - 4y' + 4y = 0 \)
• D. \( y'' - 2\sqrt{2}y' + 2y = 0 \)

Hint:

For solving differential equations given the general solution, differentiate the function as necessary and match the terms with the equation to identify the correct form.

Answer 1

The Correct Option is B

Given the solution \( y = e^{2x} \left( \cosh \sqrt{2} x + d \sinh \sqrt{2} x \right) \), we differentiate it twice to obtain the corresponding differential equation. After differentiating, we find that the correct differential equation is \( y'' - 4y' + 2y = 0 \). Thus, the correct answer is option (2).