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:
\( \sqrt{\frac{5+\sqrt{35}}{2}} \)
\( \sqrt{\frac{7+\sqrt{53}}{2}} \)
\( \frac{7+\sqrt{53}}{2} \)
\( \frac{5+\sqrt{35}}{2} \)
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 ________.
