Animated Solution for Mathematics - Differential Equations: The solution of the differential equation $\frac{dy}{dx}-\frac{y+3x}{{\log }_e(y+3x)}+3=0$ is: (where $C$ is a constant of integration.)

The given differential equation is:

At first glance, the presence of the natural logarithm in the denominator appears daunting. However, notice that the expression $(y+3x)$ appears repeatedly. In JEE Advanced problems, such repetition is a deliberate signpost indicating that a substitution will simplify the structure significantly.

To simplify, we define a new variable $t$ such that:

Differentiating this substitution with respect to $x$, we obtain:

Now, rearrange the original differential equation to isolate the term $(\frac{dy}{dx}+3)$:

By substituting our expressions for $t$ and $\frac{dt}{dx}$, the equation elegantly collapses into:

We now have a separable differential equation. Rearranging the terms to group $t$ and $x$ variables, we get:

Integrating both sides:

To solve the integral on the left, let $u={\log }_{e}t$, which implies $du=\frac{1}{t}dt$. The integral becomes $\int udu$, resulting in $\frac{u^2}{2}$. Thus, we have:

Finally, we substitute $t=y+3x$ back into our result to return to the original variables:

Rearranging this expression to a standard form, we arrive at the final solution:

The complexity of the original equation was merely a test of pattern recognition. By identifying the repeating linear term, the "monster" equation is reduced to a straightforward integration problem.