Question

Find one-parameter families of solution curves of the following differential equation:-

\[\left( x \log x \right)\frac{dy}{dx} + y = \log x\]

Solve the following differential equation:-

\[\left( x \log x \right)\frac{dy}{dx} + y = \log x\]

Answer 1

We have,
\[\left( x \log x \right)\frac{dy}{dx} + y = \log x\]
Dividing both sides by x \log x, we get
\[\frac{dy}{dx} + \frac{y}{x \log x} = \frac{\log x}{x \log x}\]
\[ \Rightarrow \frac{dy}{dx} + \frac{y}{x \log x} = \frac{1}{x}\]
\[ \Rightarrow \frac{dy}{dx} + \left( \frac{1}{x \log x} \right)y = \frac{1}{x}\]
\[\text{Comparing with }\frac{dy}{dx} + Py = Q,\text{ we get }\]
\[P = \frac{1}{x \log x} \]
\[Q = \frac{1}{x}\]
Now, 
\[I . F . = e^{\int P dx} = e^{\int\frac{1}{x \log x}dx} \]
\[ = e^{log\left( \log x \right)} \]
\[ = \log x\]
So, the solution is given by
\[y \times I.F. = \int Q \times I.F. dx + C\]
\[ \Rightarrow y \log x = \int\frac{1}{x} \times \log x\ dx + C\]
\[ \Rightarrow y \log x = \frac{\left( \log x \right)^2}{2} + C\]
\[ \Rightarrow y = \frac{1}{2}\log x + \frac{C}{\log x}\]