With initial condition $$x\left( 1 \right)\,\,\, = \,\,\,\,0.5,\,\,\,$$ the solution of the differential equation, $$\,\,\,t{{dx} \over {dt}} + x = t\,\,\,$$ is

With $$K$$ as constant, the possible solution for the first order differential equation $${{dy} \over {dx}} = {e^{ - 3x}}$$ is

The solution of the first order differential equation $$\mathop x\limits^ \bullet \left( t \right) = - 3\,x\left( t \right),\,x\left( 0 \right) = {x_0}\,\,\,\,$$ is

If at every point of a certain curve , the slope of the tangent equals $${{ - 2x} \over y},$$ the curve is _______.