The solution of the differential equation $\frac{d^2 y}{d x^2}+y=0$ is

Find the family of curves such that the angle between the tangent at any point $(x, y)$ and the tangent to the curve $xy = c^2$ at the point of intersection is $\frac{\pi}{4}$.

The foci of the curve which satisfies the differential equation $(1 + y^2) dx - xy\, dy = 0$ and passes through the point $(1, 0)$ are:

The solution of the differential equation $y dx - x dy + 3x^2 y^2 e^{x^3} dx = 0$ satisfying $y = 1$ when $x = 1$ is:

Let $y=y(x)$ be the solution of the differential equation $\frac{dy}{dx}=1+xe^{y-x}$,where $-\sqrt{2} < x < \sqrt{2}$ and $y(0)=0$. Then,the minimum value of $y(x)$ for $x \in(-\sqrt{2}, \sqrt{2})$ is equal to:

$A$ continuously differentiable function $\phi (x)$ in $(0, \pi)$ satisfying $y' = 1 + y^2$ and $y(0) = 0 = y(\pi)$ is