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Let \( x = x(y) \) be the solution of the differential equation \( y^2 \, dx + \left( x - \frac{1}{y} \right) dy = 0 \). If \( x(1) = 1 \), then \( x\!\left(\frac{1}{2}\right) \) is:

Let \( g \) be a differentiable function such that \( \int_{0}^{x} g(t)\,dt=x-\int_{0}^{x} t g(t)\,dt,\, x\ge 0 \) and let \( y=y(x) \) satisfy the differential equation \( \dfrac{dy}{dx}-y\tan x=2(x+1)\sec x\, g(x),\, x\in\left[0,\dfrac{\pi}{2}\right] \). If \( y(0)=0 \), then \( y\!\left(\dfrac{\pi}{3}\right) \) is equal to:

If \( y = y(x) \) is the solution of the differential equation, \(\displaystyle \sqrt{4 - x^2} \frac{dy}{dx} = \left( \left( \sin^{-1} \left( \frac{x}{2} \right) \right)^2 - y \right) \sin^{-1} \left( \frac{x}{2} \right) \), \( -2 \le x \le 2 \), \( \displaystyle y(2) = \frac{\pi^{2} - 8}{4} \), then \( y^{2}(0) \) is equal to _________.

Let \( f(x)=x-1 \) and \( g(x)=e^x \) for \( x \in \mathbb{R} \). If \( \frac{dy}{dx}=\Big(e^{-2\sqrt{x}} g(f(f(x))) - \frac{y}{\sqrt{x}}\Big) \), \( y(0)=0 \), then \( y(1) \) is:

Let \( y = y(x) \) be the solution of the differential equation \( \left( xy - 5x^{2}\sqrt{1 + x^{2}} \right) dx + \left( 1 + x^{2} \right) dy = 0, \ y(0) = 0 \). Then \( y(\sqrt{3}) \) is equal to:

Let \( f(x) \) be a real differentiable function such that \( f(0) = 1 \) and \( f(x+y) = f(x)f'(y) + f'(x)f(y) \) for all \( x, y \in \mathbb{R} \).
Then \( \displaystyle \sum_{n=1}^{100} \log_{e} f(n) \) is equal to:

If a curve \( y = y(x) \) passes through the point \( \left(1,\frac{\pi}{2}\right) \) and satisfies the differential equation \( \left(7x^4 \cot y - e^x cosec y\right)\frac{dy}{dx} = x^5, x \ge 1 \), then at \( x = 2 \), the value of \( \cos y \) is:

Let \( f:\mathbb{R}\rightarrow \mathbb{R} \) be a thrice differentiable odd function satisfying \( f'(x)\ge 0 \), \( f''(x)=f(x) \), \( f(0)=0 \), \( f'(0)=3 \). Then \( 9\,f(\log_{e}3) \) is equal to __________.

Let \( f \) be a differentiable function such that \( 2(x+2)^{2} f(x) - 3(x+2)^{2} = 10 \int_{0}^{x} (t+2) f(t) \, dt, \ x \ge 0 \). Then \( f(2) \) is equal to ________.

Let \( y = y(x) \) be the solution of the differential equation \( \dfrac{dy}{dx} + 2y \sec^{2} x = 2 \sec^{2} x + 3 \tan x \cdot \sec^{2} x \) such that \( y(0) = \dfrac{5}{4} \). Then \( 12 \left( y\!\left(\dfrac{\pi}{4}\right) - e^{-2} \right) \) is equal to __________________.

Let \( y=y(x) \) be the solution of the differential equation \( \dfrac{dy}{dx}+3(\tan^{2}x)\,y+3y=\sec^{2}x, \; y(0)=\dfrac{1}{3}+e^{3} \). Then \( y\!\left(\dfrac{\pi}{4}\right) \) is equal to:

Let \( y = y(x) \) be the solution of the differential equation \( (x^2+1)\,y' - 2xy = (x^4 + 2x^2 + 1)\cos x, \; y(0)=1 \).Then \(\displaystyle \int_{-3}^{3} y(x)\,dx \) is:

Let \( y = f(x) \) be the solution of the differential equation \(\displaystyle \frac{dy}{dx} + \frac{xy}{x^2 - 1} = \frac{x^6 + 4x}{\sqrt{1 - x^2}},\ -1 < x < 1 \) such that \( f(0) = 0 \). If \(\displaystyle 6 \int_{-1/2}^{1/2} f(x)\, dx = 2\pi - \alpha \) then \( \alpha^2 \) is equal to __________.