Consider the differential equation $$\,\,3y''\left( x \right) + 27y\left( x \right) = 0\,\,$$ with initial conditions $$y\left( 0 \right) = 0$$ and $$y'\left( 0 \right) = 2000.\,\,$$ The value of $$y$$ at $$x=1$$ is __________.

If $$y = f(x)$$ satiesfies the boundary value problem $$\,\,y''\,\,\, + \,\,\,9y\,\,\, = \,\,\,0,\,\,\,y\left( 0 \right)\,\,\, = \,\,\,0,\,$$ $$\,\,y\left( {{\pi \over 2}} \right) = \sqrt 2 ,\,\,\,$$ then $$\,\,y\left( {{\pi \over 4}} \right)\,\,$$ is _______.

The matrix form of the linear system $${{dx} \over {dt}} = 3x - 5y$$ and $$\,{{dy} \over {dt}} = 4x + 8y\,\,$$ is

If $$\,y = f\left( x \right)\,\,$$ is the solution of $${{{d^2}y} \over {d{x^2}}} = 0$$ with the boundary conditions $$y=5$$ at $$x=0,$$ and $$\,{{dy} \over {dx}} = 2$$ at $$x=10,$$ $$f(15)=$$_______.