y = 2 cos x + 3 sin x satisfies which of the following differential equation ?
I . `(d^(2) y)/(dx^(2)) + y = 0`
II . `((dy)/(dx))^(2) + (dy)/(dx) = 0`
Select the correct answer using the codes given below .

Options

A. Only I
B. Only II
C. Both I and II
D. Neither I nor II

Answer

To determine which differential equation the function \( y = 2 \cos x + 3 \sin x \) satisfies, we will evaluate both given equations step by step.

### Step 1: Differentiate \( y \)
We start with the function:
\[ y = 2 \cos x + 3 \sin x \]
Now, we differentiate \( y \) with respect to \( x \):
\[ \frac{dy}{dx} = \frac{d}{dx}(2 \cos x) + \frac{d}{dx}(3 \sin x) \]
Using the derivatives of sine and cosine:
\[ \frac{dy}{dx} = -2 \sin x + 3 \cos x \]

### Step 2: Differentiate \( \frac{dy}{dx} \) to find \( \frac{d^2y}{dx^2} \)
Next, we differentiate \( \frac{dy}{dx} \):
\[ \frac{d^2y}{dx^2} = \frac{d}{dx}(-2 \sin x + 3 \cos x) \]
Calculating the derivatives:
\[ \frac{d^2y}{dx^2} = -2 \cos x - 3 \sin x \]

### Step 3: Substitute \( y \) and \( \frac{d^2y}{dx^2} \) into the first differential equation
Now, we check the first differential equation:
\[ \frac{d^2y}{dx^2} + y = 0 \]
Substituting \( y \) and \( \frac{d^2y}{dx^2} \):
\[ (-2 \cos x - 3 \sin x) + (2 \cos x + 3 \sin x) = 0 \]
Simplifying:
\[ -2 \cos x - 3 \sin x + 2 \cos x + 3 \sin x = 0 \]
This simplifies to:
\[ 0 = 0 \]
Thus, the first differential equation is satisfied.

### Step 4: Check the second differential equation
Now, we check the second differential equation:
\[ \left(\frac{dy}{dx}\right)^2 + \frac{dy}{dx} = 0 \]
Substituting \( \frac{dy}{dx} \):
\[ (-2 \sin x + 3 \cos x)^2 + (-2 \sin x + 3 \cos x) = 0 \]
Let \( z = -2 \sin x + 3 \cos x \):
\[ z^2 + z = 0 \]
Factoring:
\[ z(z + 1) = 0 \]
This gives us \( z = 0 \) or \( z = -1 \). Since \( z \) is dependent on \( x \), it does not satisfy the equation for all \( x \). Therefore, the second differential equation is not satisfied.

### Conclusion
The function \( y = 2 \cos x + 3 \sin x \) satisfies the first differential equation:
\[ \frac{d^2y}{dx^2} + y = 0 \]
but does not satisfy the second one.

### Final Answer
The correct answer is **I only**.

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