The solution of the differential equation $\frac{d^2y}{dx^2} = -\frac{1}{x^2}$ is

The solution of $\frac{d^2y}{dx^2} = \sec^2 x + x e^x$ is

The curve amongst the family of curves represented by the differential equation,$(x^2 - y^2)dx + 2xy\, dy = 0$ which passes through $(1, 1)$,is

If the differential equation $\begin{vmatrix} f(x) & f'(x) \\ f'(x) & f''(x) \end{vmatrix} = 0$ holds for all $x$, with initial conditions $f(0) = 1$ and $f'(0) = 2$, then which of the following is true?

If $\phi(x) = \frac{1}{\sqrt{x}} \int \limits_0^x (4 \sqrt{2} \sin t - 3 \phi^{\prime}(t)) dt, \quad x > 0$,then $\phi^{\prime}\left(\frac{\pi}{4}\right)$ is equal to:

If $xdy = y(dx + ydy), y > 0$ and $y(1) = 1,$ then $y(-3)$ is equal to