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For \( x \in R \), let \( y(x) \) be a solution of the differential equation \( (x^{2}-5)\frac{dy}{dx}-2xy=-2x(x^{2}-5)^{2} \) such that \( y(2)=7 \). Then the maximum value of the function \( y(x) \) is:

Let \( y(x) \) be the solution of the differential equation \( x^2 \dfrac{dy}{dx} + xy = x^2 + y^2, \ x > \dfrac{1}{e} \), satisfying \( y(1) = 0 \).
Then the value of \( 2\dfrac{\,(y(e))^2}{y(e^2)} \) is _____________.

Let \( y:(-\infty,\infty)\to(0,\infty) \) be the solution of the differential equation \( \frac{dy}{dx}=\frac{e^{5x}y^{3}+y^{3}}{e^{x}+e^{x}y^{4}} \), satisfying \( y(0)=\frac{1}{\sqrt{2}} \). Then the value of \( y(\log_{e}2) \) is:

Let \( y=f(x) \) be the real valued function defined on the interval \( (0,\infty) \), satisfying \( y(1)=0 \) and the differential equation \( x\,\dfrac{dy}{dx}=y-x^3 \). Then which of the following statements is (are) TRUE?

For any real numbers \( \alpha \) and \( \beta \), let \( y_{\alpha,\beta}(x), \, x \in \mathbb{R} \), be the solution of the differential equation \( \frac{dy}{dx} + \alpha y = x e^{\beta x}, \; y(1) = 1 \).
Let \( S = \{ y_{\alpha,\beta}(x) : \alpha, \beta \in \mathbb{R} \} \). Then which of the following functions belong(s) to the set \( S \)?

For \( x\in\mathbb{R} \), let the function \( y(x) \) be the solution of the differential equation \( \dfrac{dy}{dx}+12y=\cos\!\left(\dfrac{\pi}{12}x\right),\ y(0)=0 \).
Then, which of the following statements is/are TRUE?

For all \( x>0 \), let \( y_{1}(x), y_{2}(x), \) and \( y_{3}(x) \) be the functions satisfying \( \dfrac{dy_{1}}{dx}-(\sin x)^{2}y_{1}=0, \; y_{1}(1)=5, \; \dfrac{dy_{2}}{dx}-(\cos x)^{2}y_{2}=0, \; y_{2}(1)=\dfrac{1}{3}, \; \dfrac{dy_{3}}{dx}-\left(\dfrac{2-x^{3}}{x^{3}}\right)y_{3}=0, \; y_{3}(1)=\dfrac{3}{5e}, \) respectively. Then \( \displaystyle \lim_{x\to 0^{+}} \dfrac{y_{1}(x)y_{2}(x)y_{3}(x)+2x}{e^{3x}\sin x} \) is equal to________.

Let \( f : [1, \infty) \rightarrow \mathbb{R} \) be a differentiable function such that \( f(1) = \frac{1}{3} \) and\( 3 \int_{1}^{x} f(t)\, dt = x f(x) - \frac{x^{3}}{3} \), \( x \in [1, \infty) \).
Let \( e \) denote the base of the natural logarithm. Then the value of \( f(e) \) is:

If \( y(x) \) is the solution of the differential equation \( x\,dy-(y^2-4y)\,dx=0 \) for \( x>0 \), \( y(1)=2 \), and the slope of the curve \( y=y(x) \) is never zero, then the value of \( 10y(\sqrt{2}) \) is ________.

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JEE Advance 2022 Paper -2

If \( y(x) \) is the solution of the differential equation \( x\,dy-(y^2-4y)\,dx=0 \) for \( x>0 \), \( y(1)=2 \), and the slope of the curve \( y=y(x) \) is never zero, then the value of \( 10y(\sqrt{2}) \) is ________.