Given that $$\mathop x\limits^{ \bullet \bullet } + 3x = 0$$ and $$x\left( 0 \right) = 1,\,\,\mathop x\limits^ \bullet \left( 0 \right) = 1,$$ What is $$x(1)$$ ________.

The solution of $${{d\,y} \over {d\,x}} = {y^2}$$ with initial value $$y(0)=1$$ is bounded in the interval is

The partial differential equation $$\,\,{{{\partial ^2}\phi } \over {\partial {x^2}}} + {{{\partial ^2}\phi } \over {\partial {y^2}}} + {{\partial \phi } \over {\partial x}} + {{\partial \phi } \over {\partial y}} = 0\,\,\,\,$$ has

For $$\,\,\,{{{d^2}y} \over {d{x^2}}} + 4{{dy} \over {dx}} + 3y = 3{e^{2x}},\,\,$$ the particular integral is