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?
The function \( f \) has a local minimum at \( x=\dfrac{1}{\sqrt{3}} \)
The function \( f \) has a local maximum at \( x=\dfrac{1}{\sqrt{3}} \)
The function \( f \) is increasing in the interval \( (1,\,2) \)
If \( g(x)=4x^3-5x^2+\dfrac{3}{2}x \) for \( x>0 \), then the number of elements in the set \( \{\,x\in(0,\infty):\,f(x)=g(x)\,\} \) is \( 2 \)
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 ________.
