Question

If \(y = e^{-mx}\) is a solution of the differential equation \(\frac{d^2y}{dx^2}+4\frac{dy}{dx}+3y = 0\), then the values of \(m\) are

• A. \(1,3\)
• B. \(-1,3\)
• C. \(-1,-3\)
• D. \(1,-3\)

Hint:

Substitute \(y=e^{-mx}\) and solve for \(m\).

Answer 1

The Correct Option is A

Step 1: Understanding the Concept
If \(y=e^{-mx}\) then \(y'=-me^{-mx}\) and \(y''=m^2e^{-mx}\).

Step 2: Key Formula or Approach
Substitute into \(y''+4y'+3y=0\).

Step 3: Detailed Explanation
\[ m^2e^{-mx}-4me^{-mx}+3e^{-mx}=0 \Rightarrow m^2-4m+3=0 \]
\[ (m-1)(m-3)=0 \Rightarrow m=1,\,3 \]

Final Answer:
The values of \(m\) are 1 and 3, option (A). \[ \boxed{m=1,\ 3\ \text{(A)}} \]