If ` y= 3e^(2x)+ 2e^(3x) ,then (d^(2)y)/(dx^(2))-5(dy)/(dx) =`

Options

A. 6y
B. `-6y `
C. ` 3y `
D. ` -3y `

Answer

To solve the problem, we need to find the second derivative of the function \( y = 3e^{2x} + 2e^{3x} \) and then subtract \( 5 \) times the first derivative from it.

### Step 1: Find the first derivative \( \frac{dy}{dx} \)

Given:
\[ y = 3e^{2x} + 2e^{3x} \]

Using the chain rule, we differentiate each term:
- The derivative of \( 3e^{2x} \) is \( 3 \cdot e^{2x} \cdot 2 = 6e^{2x} \)
- The derivative of \( 2e^{3x} \) is \( 2 \cdot e^{3x} \cdot 3 = 6e^{3x} \)

Thus, the first derivative is:
\[ \frac{dy}{dx} = 6e^{2x} + 6e^{3x} \]

### Step 2: Find the second derivative \( \frac{d^2y}{dx^2} \)

Now we differentiate \( \frac{dy}{dx} \):
\[ \frac{dy}{dx} = 6e^{2x} + 6e^{3x} \]

Differentiating each term:
- The derivative of \( 6e^{2x} \) is \( 6 \cdot e^{2x} \cdot 2 = 12e^{2x} \)
- The derivative of \( 6e^{3x} \) is \( 6 \cdot e^{3x} \cdot 3 = 18e^{3x} \)

Thus, the second derivative is:
\[ \frac{d^2y}{dx^2} = 12e^{2x} + 18e^{3x} \]

### Step 3: Substitute into the expression \( \frac{d^2y}{dx^2} - 5 \frac{dy}{dx} \)

Now we substitute \( \frac{d^2y}{dx^2} \) and \( \frac{dy}{dx} \) into the expression:
\[ \frac{d^2y}{dx^2} - 5 \frac{dy}{dx} = (12e^{2x} + 18e^{3x}) - 5(6e^{2x} + 6e^{3x}) \]

Calculating \( -5 \frac{dy}{dx} \):
\[ -5(6e^{2x} + 6e^{3x}) = -30e^{2x} - 30e^{3x} \]

Now combine the terms:
\[ \frac{d^2y}{dx^2} - 5 \frac{dy}{dx} = (12e^{2x} + 18e^{3x}) - (30e^{2x} + 30e^{3x}) \]
\[ = (12e^{2x} - 30e^{2x}) + (18e^{3x} - 30e^{3x}) \]
\[ = -18e^{2x} - 12e^{3x} \]

### Final Answer
\[ \frac{d^2y}{dx^2} - 5 \frac{dy}{dx} = -18e^{2x} - 12e^{3x} \]