Animated Solution for Mathematics - Differential Equations: A solution of the differential equation $(\frac{dy}{dx}{)}^2-x\frac{dy}{dx}+y=0$ is

Imagine you are standing before a differential equation that looks like a tangled knot:

At first glance, the squared derivative ${\left(\frac{dy}{dx}\right)}^2$ might send a shiver down your spine. It feels non-linear, perhaps even chaotic.

But in the world of JEE Advanced, appearances are often deceptive. The secret to solving this isn't brute force; it's perspective.

By moving everything except $y$ to the other side, we get:

Suddenly, the chaos settles. We have $y$ expressed as a function of $x$ and its first derivative. This is the first step in our journey.

Now, let's simplify our notation. Let $p=\frac{dy}{dx}$. The equation transforms into:

Does this look familiar? It should! This is the hallmark of a Clairaut's Equation, which takes the general form $y=xp+f(p)$.

In our case, $f(p)=-p^2$. Recognizing this form is like finding a hidden door in a maze.

It tells us that we don't need to struggle with complex integration techniques. The mathematician Alexis Clairaut gave us a beautiful gift: for any equation of this form, the general solution is simply $y=cx+f(c)$, where $c$ is an arbitrary constant.

Applying this rule is almost too easy. We replace $p$ with $c$ and $f(p)$ with $f(c)$.

Since $f(p)=-p^2$, it follows that $f(c)=-c^2$. Therefore, our general solution is:

Geometrically, this represents a family of straight lines. Each value of $c$ corresponds to a unique line in the Cartesian plane.

We have moved from a terrifying non-linear differential equation to a simple family of lines. This is the power of pattern recognition in mathematics.

To be absolutely certain, let's test the option $y=2x-4$. If we compare this to our general solution $y=cx-c^2$, we see that if we set $c=2$, then $c^2=4$, which perfectly matches our option.

To verify, we differentiate $y=2x-4$ to get $\frac{dy}{dx}=2$. Substituting $y=2x-4$ and $\frac{dy}{dx}=2$ back into the original equation:

The left-hand side equals the right-hand side. The solution is confirmed!

We have successfully navigated the problem. The key takeaway is to always keep an eye out for Clairaut's form.

Whenever you see an equation that can be written as $y=xp+f(p)$, remember the shortcut: $y=cx+f(c)$.

It is a tool that will save you precious time and boost your confidence in the exam hall. Keep practicing, keep visualizing, and most importantly, keep falling in love with the elegance of mathematics.