The maximum value of the solution $$y$$ $$(t)$$ of the differential equation $$\,\,y\left( t \right) + \mathop y\limits^{ \bullet \,\, \bullet } \left( t \right) = 0\,\,\,$$ with initial conditions $$\,\,\mathop y\limits^ \bullet \left( 0 \right) = 1\,\,$$ and $$\,\,y\left( 0 \right) = 1,\,\,$$ for $$\,t \ge 0\,\,$$ is

The type of the partial differential equation $${{\partial f} \over {\partial t}} = {{{\partial ^2}f} \over {\partial {x^2}}}\,\,is$$

While numerically solving the differential equation $$\,{{dy} \over {dx}} + 2x{y^2} = 0,y\left( 0 \right) = 1\,\,$$ using Euler's predictor corrector (improved Euler- Cauchy) method with a step size of $$0.2,$$ the value of $$y$$ after the first step is