y = 4sin 3x is a solution of the differential equation

Options

A. `(dy)/(dx)+ 8y=0`
B. `(dy)/(dx)- 8y=0`
C. `(d^2y)/(dx^2)+ 9y =0`
D. `(d^2y)/(dx^2)- 9y =0`

Answer

To show that \( y = 4 \sin(3x) \) is a solution of a differential equation, we will differentiate it twice and then substitute it into the equation to find the corresponding differential equation.

### Step 1: Differentiate \( y \) with respect to \( x \)

Given:
\[ y = 4 \sin(3x) \]

We differentiate \( y \):
\[ \frac{dy}{dx} = 4 \cdot \cos(3x) \cdot \frac{d}{dx}(3x) = 4 \cdot \cos(3x) \cdot 3 = 12 \cos(3x) \]

### Step 2: Differentiate \( \frac{dy}{dx} \) to find \( \frac{d^2y}{dx^2} \)

Now, we differentiate \( \frac{dy}{dx} \):
\[ \frac{d^2y}{dx^2} = \frac{d}{dx}(12 \cos(3x)) = 12 \cdot (-\sin(3x)) \cdot \frac{d}{dx}(3x) = 12 \cdot (-\sin(3x)) \cdot 3 = -36 \sin(3x) \]

### Step 3: Substitute \( y \) and \( \frac{d^2y}{dx^2} \) into the differential equation

We have:
\[ \frac{d^2y}{dx^2} = -36 \sin(3x) \quad \text{and} \quad y = 4 \sin(3x) \]

Now, we can express \( \frac{d^2y}{dx^2} + 9y = 0 \):
\[ \frac{d^2y}{dx^2} + 9y = -36 \sin(3x) + 9(4 \sin(3x)) = -36 \sin(3x) + 36 \sin(3x) = 0 \]

### Conclusion

Thus, we have shown that:
\[ \frac{d^2y}{dx^2} + 9y = 0 \]
is the differential equation for which \( y = 4 \sin(3x) \) is a solution.

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