Question

Let \( y = y(x) \) be the solution of the differential equation \[\left( 2x \log_e x \right) \frac{dy}{dx} + 2y = \frac{3}{x} \log_e x, \, x>0 \, \text{and} \, y(e^{-1}) = 0.\] Then, \( y(e) \) is equal to:

• A. \( \frac{3}{2e} \)
• B. \( \frac{-2}{3e} \)
• C. \( \frac{-3}{e} \)
• D. \( \frac{-2}{e} \)

Answer 1

The Correct Option is C

The differential equation to be solved is:

\[ \frac{dy}{dx} + \frac{y}{x \ln x} = \frac{3}{2x^2}. \]

Step 1: Calculate the integrating factor (I.F.).

\[ \text{I.F.} = e^{\int \frac{1}{x \ln x} dx} = e^{\ln(\ln x)} = \ln x. \]

Step 2: Multiply the equation by the I.F.

\[ (\ln x)y = \int \frac{3 \ln x}{2x^2} dx. \]

Step 3: Evaluate the integral on the right side.

\[ \int \frac{3 \ln x}{2x^2} dx = \frac{3}{2} \int x^{-2} \ln x dx. \]

Employ integration by parts with \( u = \ln x \) and \( dv = x^{-2} dx \):

\[ \int \ln x \cdot x^{-2} dx = -\frac{\ln x}{x} - \int -\frac{1}{x^2} dx = -\frac{\ln x}{x} + \frac{1}{x}. \]

The integral becomes:

\[ \int \frac{3 \ln x}{2x^2} dx = \frac{3}{2} \left( -\frac{\ln x}{x} + \frac{1}{x} \right). \]

Step 4: Formulate the general solution.

\[ y \ln x = \frac{3}{2} \left( -\frac{\ln x}{x} + \frac{1}{x} \right) + C. \]

Simplify the expression for \(y\):

\[ y = -\frac{3 \ln x}{2x \ln x} + \frac{3}{2x \ln x} + \frac{C}{\ln x}. \]

\[ y = -\frac{3}{2x} + \frac{C}{\ln x}. \]

Step 5: Apply the initial condition \( y(e^{-1}) = 0 \) to find \(C\).

\[ 0 = -\frac{3}{2e^{-1}} + \frac{C}{\ln(e^{-1})}. \]

\[ 0 = -\frac{3}{2e} + \frac{C}{-1}. \]

Solving for \(C\) yields:

\[ C = -\frac{3}{2}e. \]

Step 6: Determine \( y(e) \) using the calculated value of \(C\).

\[ y(e) = -\frac{3}{2e} + \frac{-\frac{3}{2}e}{\ln e}. \]

\[ y(e) = -\frac{3}{2e} - \frac{3}{2e} = -\frac{3}{e}. \]

Final Answer: \(-\frac{3}{e}\).