Animated Solution for Mathematics - Differential Equations: Let $y=y(x)$ be the solution of the differential equation $\sin x\frac{dy}{dx}+y\cos x=4x,x\in (0,\pi )$. If $y(\pi /2)=0$, then $y(\pi /6)$ is equal to :

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

At first glance, it feels like a standard linear differential equation, and your instinct might be to reach for the integrating factor method. But wait! Before you dive into the deep end of exponential integrals, pause.

Look at the left-hand side. There is a hidden symmetry here, a mathematical elegance waiting to be uncovered. In the world of JEE Advanced, recognizing these patterns is the difference between a frantic, time-consuming calculation and a swift, elegant victory.

Let us look at the expression $\sin x\frac{dy}{dx}+y\cos x$. Does it ring a bell?

Recall the product rule of differentiation:

If we assign $u=y$ and $v=\sin x$, the derivative becomes $y\frac{d}{dx}(\sin x)+\sin x\frac{dy}{dx}$. Since the derivative of $\sin x$ is $\cos x$, this simplifies perfectly to $y\cos x+\sin x\frac{dy}{dx}$.

It is a perfect match! The entire left-hand side of our equation is simply the derivative of the product $y\sin x$. We can rewrite our intimidating equation as:

Suddenly, the complexity vanishes. We have reduced a differential equation to a simple derivative equality.

Now that we have $\frac{d}{dx}(y\sin x)=4x$, the path forward is illuminated. We integrate both sides with respect to $x$:

By the Fundamental Theorem of Calculus, the integral of a derivative is the function itself. Thus, the left side becomes $y\sin x$. On the right side, the integral of $4x$ is $2x^2$.

But we must never forget the constant of integration, $C$. Our general solution is:

This constant $C$ is the anchor of our solution; it represents the infinite family of curves that satisfy this differential equation.

We are given the initial condition $y(\frac{\pi }{2})=0$. This is our key to unlocking the specific curve we need.

We substitute $x=\frac{\pi }{2}$ and $y=0$ into our general solution:

Since $\sin (\frac{\pi }{2})=1$, the left side is zero. The right side becomes $2(\frac{{\pi }^2}{4})+C$, which simplifies to $\frac{{\pi }^2}{2}+C$. Therefore, $C=-\frac{{\pi }^2}{2}$.

We have found our anchor. The particular solution is:

Our goal is to find $y(\frac{\pi }{6})$. We substitute $x=\frac{\pi }{6}$ into our particular solution:

We know that $\sin (\frac{\pi }{6})=\frac{1}{2}$ and $(\frac{\pi }{6}{)}^2=\frac{{\pi }^2}{36}$. Substituting these values, we get:

This simplifies to $y(\frac{\pi }{6})\cdot \frac{1}{2}=\frac{{\pi }^2}{18}-\frac{{\pi }^2}{2}$. To subtract these, we use a common denominator of $18$:

Finally, multiplying both sides by $2$, we arrive at:

We have navigated the complexity, respected the rules of calculus, and arrived at the solution with precision. This is the beauty of mathematics—when you look past the initial intimidation, you find a logical, harmonious structure waiting to be solved.