Animated Solution for Mathematics - Differential Equations: If $y=y(x)$ satisfies the differential equation $16\sqrt{x+9\sqrt{x}}(4+\sqrt{9+\sqrt{x}})\cos ydy=(1+2\sin y)dx,x>0$ and $y(256)=\frac{\pi }{2},y(49)=\alpha$, then $2\sin \alpha$ is equal to :

Welcome, future engineer. Today, we are going to stare down a problem that, at first glance, looks like it was designed to break your spirit. We have a differential equation: $16\sqrt{x+9\sqrt{x}}(4+\sqrt{9+\sqrt{x}})\cos ydy=(1+2\sin y)dx$.

It is messy, it is nested, and it is intimidating. But here is the secret of the JEE Advanced: the more complex a problem looks, the more likely it is that there is a beautiful, elegant symmetry hidden just beneath the surface. Our job is not to fight the complexity, but to peel it back layer by layer.

The Art of Separation

In the world of differential equations, our first instinct should always be to bring order to chaos. We use the Variable Separable Method. We want all the $y$ terms on the left and all the $x$ terms on the right.

By rearranging our equation, we get:

Look at that. The chaos has subsided. We have separated the variables. Now, we can integrate both sides independently.

Taming the Left Hand Side

Let us look at the left side: $\int \frac{\cos y}{1+2\sin y}dy$. This is a classic substitution problem. If you look closely, the numerator $\cos y$ is proportional to the derivative of the denominator $1+2\sin y$.

Let $u=1+2\sin y$. Then $du=2\cos ydy$, or $\cos ydy=\frac{du}{2}$. Substituting this in, we get:

Simple, elegant, and clean. We have conquered the left side.

The Hidden Symmetry of the Right Hand Side

Now, for the right side. This is where most students panic. The expression $\int \frac{dx}{16\sqrt{x+9\sqrt{x}}(4+\sqrt{9+\sqrt{x}})}$ looks like a labyrinth.

Let us define $t=4+\sqrt{9+\sqrt{x}}$. We need to find $dt$ by differentiating $t$ with respect to $x$ using the chain rule.

Look at that result! It matches the denominator of our integral perfectly. Substituting $dt$ into our integral, we get:

It is not a nightmare; it is a masterpiece of design. We have reduced a terrifying expression into a manageable logarithmic form.

The Final Convergence

Now we combine our results:

We are given the boundary condition $y(256)=\frac{\pi }{2}$. Let us plug in $x=256$ and $y=\frac{\pi }{2}$ to find our constant $C$.

Since $\sin (\frac{\pi }{2})=1$, the left side becomes $\frac{1}{2}\ln (3)$. On the right side, $\sqrt{256}=16$, $9+16=25$, $\sqrt{25}=5$, and $4+5=9$. Thus, we get $\frac{1}{4}\ln (9)=\frac{1}{4}\ln (3^2)=\frac{1}{2}\ln (3)$.

Equating them: $\frac{1}{2}\ln (3)=\frac{1}{2}\ln (3)+C$. This implies $C=0$.

The Victory Lap

Finally, we use the second condition: $y(49)=\alpha$. Plugging in $x=49$:

Simplifying the right side: $\sqrt{49}=7$, $9+7=16$, $\sqrt{16}=4$, and $4+4=8$. We have $\frac{1}{4}\ln (8)$, which simplifies to $\frac{1}{2}\ln (\sqrt{8})=\frac{1}{2}\ln (2\sqrt{2})$.

Equating the arguments of the logarithms:

And there it is. The monster is defeated. You didn't just solve a problem; you navigated a complex landscape of calculus and emerged victorious.