Question

Find the general solution of the differential equation : \( y \log y \frac{dx}{dy} + x = \frac{2}{y} \).

Hint:

Notice that the log term in \( Q(y) \) cancels out with the \( IF \), simplifying the final integration.

Answer 1

Concept:
• Linear differential equation in \( x \): \( \frac{dx}{dy} + P(y)x = Q(y) \).
• Integrating Factor \( IF = e^{\int P(y) dy} \).

Step 1: Identify standard form
Divide by \( y \log y \):
\[ \frac{dx}{dy} + \frac{1}{y \log y} x = \frac{2}{y^2 \log y} \]
\( P(y) = \frac{1}{y \log y} \) and \( Q(y) = \frac{2}{y^2 \log y} \).

Step 2: Calculate IF
\[ \int P(y) dy = \int \frac{1/y}{\log y} dy = \log(\log y) \]
\[ IF = e^{\log(\log y)} = \log y \]

Step 3: Solve the equation
\[ x \cdot IF = \int Q(y) \cdot IF \, dy + C \]
\[ x \log y = \int \frac{2}{y^2 \log y} \cdot \log y \, dy = \int \frac{2}{y^2} dy \]
\[ x \log y = -\frac{2}{y} + C \]