General solution of differential equation $y \log y dx - x dy = 0$ is :

Step 1: Separate the variables

The given differential equation is $y \log y dx - x dy = 0$. To solve this, we first rearrange the terms to separate the variables $x$ and $y$. We move the $x dy$ term to the right side and then divide both sides by $x$ and $y \log y$ to group $dx$ with $x$ and $dy$ with $y$.

Step 2: Integrate both sides

Now that the variables are separated, we integrate both sides of the equation. The left side is a standard integral of $1/x$, and the right side requires a substitution to solve.

Step 3: Solve the integrals

For the left side, $\int \frac{1}{x} dx = \log |x|$. For the right side, let $u = \log y$. Then $du = \frac{1}{y} dy$. So, $\int \frac{1}{y \log y} dy = \int \frac{1}{u} du = \log |u| = \log |\log y|$. We add the constant of integration $C$ to one side.

Step 4: Simplify the general solution

We can combine the logarithmic terms on the left side using the property $\log a - \log b = \log (a/b)$. Then, we convert the logarithmic equation to an exponential form. Let $e^C = A$, where $A$ is an arbitrary positive constant. This gives us the general solution.

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