Animated Solution for Mathematics - Differential Equations: If $y=y(x)$ is the solution of the differential equation $\frac{dy}{dx}+(\tan x)y=\sin x,0\le x\le \frac{\pi }{3}$, with $y(0)=0$, then $y\left(\frac{\pi }{4}\right)$ equal to :

Welcome, fellow traveler on the path of JEE Advanced mastery! Today, we are not just solving a differential equation; we are uncovering the hidden geometry of a function.

The problem asks us to solve:

with the initial condition $y(0)=0$. It might look like a jumble of trigonometric functions, but beneath the surface lies a beautiful, structured symmetry.

Recognizing the Architecture

Whenever you see a differential equation, your first instinct should be to check its form. Is it separable? Is it homogeneous? Or is it linear?

Our equation fits the classic mold of a First-order Linear Differential Equation:

Here, our $P(x)=\tan x$ and our $Q(x)=\sin x$. Recognizing this is like finding the map to the treasure; it tells us that we have a reliable, systematic way to reach the solution: the Integrating Factor method.

The Magic Multiplier

To solve this, we need an Integrating Factor (I.F.). Think of the I.F. as a catalyst in a chemical reaction—it doesn't change the nature of the equation, but it makes the integration possible.

The formula is:

Substituting our $P(x)$, we get:

Because the exponential and natural logarithm are inverse functions, they cancel out, leaving us with a beautifully simple $\text{I.F.}=\sec x$.

The Collapse of Complexity

Now, we multiply our entire original equation by $\sec x$. The left side becomes:

If you look closely, the left side is exactly the derivative of the product $y\cdot \sec x$. This is the 'collapse' I mentioned—the complexity vanishes, leaving us with:

Integrating both sides with respect to $x$, we get:

The Anchor and the Final Leap

We are almost at the finish line. We have a family of curves, but we need the one that passes through the origin.

Using the condition $y(0)=0$:

Our particular solution is $y=\cos x\ln |\sec x|$.

Finally, we evaluate this at $x=\frac{\pi }{4}$. We know $\cos (\frac{\pi }{4})=\frac{1}{\sqrt{2}}$ and $\sec (\frac{\pi }{4})=\sqrt{2}$.

Plugging these in:

Using the property $\ln (a^b)=b\ln a$, we simplify $\ln (\sqrt{2})$ to $\frac{1}{2}\ln 2$. The final result is:

Isn't it satisfying? We started with a complex differential equation and, through the logic of the integrating factor, arrived at a clean, precise value. Keep practicing this flow, and you will find that even the most intimidating equations become stories waiting to be told.