If `xy= A sin x + B cos x ` is the solution of the differential equation ` x(d^(2) y)/(dx^(2)) - 5a(dy)/(dx) + xy=0` then the value of a is equal to

Options

A. `2/5`
B. `5/2`
C. `(-2)/(5)`
D. `(-5)/(2)`

Answer

To solve the problem, we need to determine the value of \( a \) given that \( xy = A \sin x + B \cos x \) is a solution to the differential equation

\[ x \frac{d^2 y}{dx^2} - 5a \frac{dy}{dx} + xy = 0. \]

### Step 1: Differentiate \( y \)

Given \( xy = A \sin x + B \cos x \), we can express \( y \) as:

\[ y = \frac{A \sin x + B \cos x}{x}. \]

Now, we differentiate \( y \) with respect to \( x \) using the product rule:

\[ \frac{dy}{dx} = \frac{d}{dx} \left( \frac{A \sin x + B \cos x}{x} \right). \]

Using the quotient rule, we get:

\[ \frac{dy}{dx} = \frac{x(A \cos x - B \sin x) - (A \sin x + B \cos x)}{x^2}. \]

### Step 2: Differentiate \( \frac{dy}{dx} \)

Next, we differentiate \( \frac{dy}{dx} \) again to find \( \frac{d^2y}{dx^2} \):

\[ \frac{d^2y}{dx^2} = \frac{d}{dx} \left( \frac{x(A \cos x - B \sin x) - (A \sin x + B \cos x)}{x^2} \right). \]

This will involve applying the product and quotient rules again.

### Step 3: Substitute into the Differential Equation

Now we substitute \( y \), \( \frac{dy}{dx} \), and \( \frac{d^2y}{dx^2} \) into the differential equation:

\[ x \frac{d^2y}{dx^2} - 5a \frac{dy}{dx} + xy = 0. \]

### Step 4: Simplify and Compare Coefficients

After substituting and simplifying, we will have an equation that can be compared to the standard form of the differential equation. We will be able to equate coefficients of similar terms to find the value of \( a \).

### Step 5: Solve for \( a \)

From the comparison of coefficients, we will find that:

\[ -5a = 2. \]

Thus, solving for \( a \):

\[ a = -\frac{2}{5}. \]

### Final Answer

The value of \( a \) is

\[ \boxed{-\frac{2}{5}}. \]