Question

The solution of the differential equation \(\frac{d^2x}{dt^2}+6\frac{dx}{dt}+9x=0\) is given by:

• A. $\text{x} = (\text{C}_1 + \text{C}_2\text{t})\text{e}^{3\text{t}}$
• B. $\text{x} = (\text{C}_1 + \text{C}_2\text{t})\text{e}^{-3\text{t}}$
• C. $\text{x} = (\text{C}_1 + \text{C}_2\text{t})\text{e}^{5\text{t}}$
• D. $\text{x} = (\text{C}_1 + \text{C}_2\text{t})\text{e}^{-5\text{t}}$

Hint:

For repeated roots, always multiply the second constant by the independent variable ($t$ or $x$) to maintain linear independence of the solutions: $x = (C_1 + C_2 t)e^{mt}$.

Answer 1

The Correct Option is B

Step 1: Understanding the Concept:
This is a homogeneous linear second-order ordinary differential equation with constant coefficients.
To solve it, we find the roots of its auxiliary (characteristic) equation.
Key Formula or Approach:
For an equation of the form $a \frac{d^2x}{dt^2} + b \frac{dx}{dt} + cx = 0$, the auxiliary equation is: \[ a m^2 + b m + c = 0 \] If the roots are real and equal ($m_1 = m_2 = m$), the general solution is: \[ x = (C_1 + C_2 t) e^{mt} \]

Step 2: Detailed Explanation:
The given differential equation is: \[ \frac{d^2x}{dt^2} + 6 \frac{dx}{dt} + 9x = 0 \] Its corresponding auxiliary equation is: \[ m^2 + 6m + 9 = 0 \] Factor the quadratic equation: \[ (m + 3)^2 = 0 \] Solving for $m$ yields equal roots: \[ m_1 = m_2 = -3 \] Since the roots are real and repeated, the general solution is: \[ x = (C_1 + C_2 t) e^{-3t} \]

Step 3: Final Answer:
The solution matches Option (B).