Let `y=(A+Bx)e^(3x)` is a Solution of the differential equation `(d^(2)y)/(dx^(2))+m(dy)/(dx)+ny=0,m,n in I,` then

Options

A. m=-6
B. n=-6
C. m=9
D. n=9

Answer

To solve the problem, we need to find the values of \( m \) and \( n \) in the differential equation given that \( y = (A + Bx)e^{3x} \) is a solution. We will differentiate \( y \) twice and then compare the resulting differential equation with the standard form provided.

### Step 1: Define the function
Let:
\[ y = (A + Bx)e^{3x} \]

### Step 2: Differentiate \( y \) once
Using the product rule:
\[ \frac{dy}{dx} = \frac{d}{dx}[(A + Bx)e^{3x}] = (A + Bx) \frac{d}{dx}[e^{3x}] + e^{3x} \frac{d}{dx}[A + Bx] \]
Calculating the derivatives:
\[ \frac{d}{dx}[e^{3x}] = 3e^{3x}, \quad \frac{d}{dx}[A + Bx] = B \]
Thus,
\[ \frac{dy}{dx} = (A + Bx)(3e^{3x}) + e^{3x}(B) = 3(A + Bx)e^{3x} + Be^{3x} \]
Combining terms:
\[ \frac{dy}{dx} = (3A + 3Bx + B)e^{3x} = (3A + B + 3Bx)e^{3x} \]

### Step 3: Differentiate \( y \) again
Now, we differentiate \( \frac{dy}{dx} \):
\[ \frac{d^2y}{dx^2} = \frac{d}{dx}[(3A + B + 3Bx)e^{3x}] \]
Applying the product rule again:
\[ \frac{d^2y}{dx^2} = (3A + B + 3Bx) \frac{d}{dx}[e^{3x}] + e^{3x} \frac{d}{dx}[3A + B + 3Bx] \]
Calculating the derivatives:
\[ \frac{d}{dx}[3A + B + 3Bx] = 3B \]
Thus,
\[ \frac{d^2y}{dx^2} = (3A + B + 3Bx)(3e^{3x}) + e^{3x}(3B) = (3A + B + 3Bx)(3)e^{3x} + 3Be^{3x} \]
Combining terms:
\[ \frac{d^2y}{dx^2} = (9A + 3B + 9Bx + 3B)e^{3x} = (9A + 6B + 9Bx)e^{3x} \]

### Step 4: Form the differential equation
Now we have:
\[ \frac{d^2y}{dx^2} = (9A + 6B + 9Bx)e^{3x} \]
We can express the second derivative in terms of \( y \):
\[ \frac{d^2y}{dx^2} - 6\frac{dy}{dx} + 9y = 0 \]
Substituting \( y \) and \( \frac{dy}{dx} \):
\[ (9A + 6B + 9Bx)e^{3x} - 6(3A + B + 3Bx)e^{3x} + 9(A + Bx)e^{3x} = 0 \]

### Step 5: Compare coefficients
The resulting equation simplifies to:
\[ (9A + 6B + 9Bx - 18A - 6B - 18Bx + 9A + 9Bx)e^{3x} = 0 \]
This leads to:
\[ 0 = 0 \]
Thus, we can identify:
\[ m = -6, \quad n = 9 \]

### Final Answer
The values of \( m \) and \( n \) are:
\[ m = -6, \quad n = 9 \]