Question

Let \(p\) and \(q\) be fixed real numbers. If the function \(y(x) = C_1 e^{3x} + C_2 e^{-4x}\) is a solution of the differential equation \(\frac{d^2 y}{dx^2} + p\frac{dy}{dx} + qy = 0\); for any constants \(C_1\) and \(C_2\), then \(p - q\) is equal to _______

• A. \(-1\)
• B. 7
• C. \(-12\)
• D. 13

Hint:

Reconstruct the characteristic equation directly as \((r - r_1)(r - r_2) = 0\).
This immediately gives the coefficients: the coefficient of the single-derivative term is the negative sum of the roots, and the constant term is the product of the roots.

Answer 1

The Correct Option is D

Step 1: Understanding the Question:
The question provides the general solution \(y(x) = C_1 e^{3x} + C_2 e^{-4x}\) to a linear second-order homogeneous ordinary differential equation with constant coefficients.
We need to determine the constant parameters \(p\) and \(q\) of the differential equation and then calculate the difference \(p - q\).

Step 2: Key Formula or Approach:
For a second-order homogeneous linear differential equation with constant coefficients:
\[ \frac{d^2 y}{dx^2} + p\frac{dy}{dx} + qy = 0 \] The characteristic equation is obtained by substituting \(y = e^{rx}\):
\[ r^2 + pr + q = 0 \] The roots \(r_1\) and \(r_2\) of this characteristic equation determine the exponents in the general solution:
\[ y(x) = C_1 e^{r_1 x} + C_2 e^{r_2 x} \] Using the relationships between the roots and coefficients of a quadratic equation:
- Sum of roots: \(r_1 + r_2 = -p\)
- Product of roots: \(r_1 \cdot r_2 = q\)

Step 3: Detailed Explanation:
Let us extract the roots from the given general solution:
The solution is \(y(x) = C_1 e^{3x} + C_2 e^{-4x}\).
By comparing this with the standard solution form, the roots of the characteristic equation are:
\[ r_1 = 3 \] \[ r_2 = -4 \] Now we can find the coefficients \(p\) and \(q\) using the root relationships:
1. Calculate \(p\) from the sum of the roots:
\[ r_1 + r_2 = 3 + (-4) = -1 \] Since \(r_1 + r_2 = -p\):
\[ -p = -1 \implies p = 1 \] 2. Calculate \(q\) from the product of the roots:
\[ q = r_1 \cdot r_2 = 3 \times (-4) = -12 \] 3. Alternatively, we can reconstruct the characteristic equation directly from its factors:
\[ (r - r_1)(r - r_2) = 0 \] \[ (r - 3)(r + 4) = 0 \] \[ r^2 + 4r - 3r - 12 = 0 \] \[ r^2 + r - 12 = 0 \] Comparing this to \(r^2 + pr + q = 0\), we confirm that \(p = 1\) and \(q = -12\).
4. Calculate the required difference \(p - q\):
\[ p - q = 1 - (-12) = 1 + 12 = 13 \] This matches Option (D) exactly.

Step 4: Final Answer
Thus, the value of \(p - q\) is \(13\), which corresponds to option (D).