Consider the differential equation $\dot{\vec{w}}=A \vec{w}$, with $\vec{w}(t=0)=\left[\begin{array}{l}1 \\ 1\end{array}\right]$.

If $\vec{w}(t)=e^t \vec{u}_x+e^{-2 t} \vec{u}_y$ be the solution to the equation where $\vec{u}_x$ and $\vec{u}_y$ are unit vectors along the positive x and y axes respectively, then which of the following options is the correct matrix representing $A$ ?

The function $y(t)$ satisfies

$$ t^2 y^{\prime \prime}(t)-2 t y^{\prime}(t)+2 y(t)=0 $$

where $y^{\prime}(t)$ and $y^{\prime \prime}(t)$ denote the first and second derivatives of $y(t)$, respectively. Given $y^{\prime}(0)=1$ and $y^{\prime}(1)=-1$, the maximum value of $y(t)$ over $[0,1]$ is ___________ (rounded off to two decimal places).

The general form of the complementary function of a differential equation is given by $y(t) = (At + B)e^{-2t}$, where $A$ and $B$ are real constants determined by the initial condition. The corresponding differential equation is ____.

Consider the following partial differential equation (PDE)

$$a{{{\partial ^2}f(x,y)} \over {\partial {x^2}}} + b{{{\partial ^2}f(x,y)} \over {\partial {y^2}}} = f(x,y)$$,

where a and b are distinct positive real numbers. Select the combination(s) of values of the real parameters $$\xi $$ and $$\eta $$ such that $$f(x,y) = {e^{\xi x + \eta y}}$$ is a solution of the given PDE.