The solution of the differential equation $$\,{x^2}{{dy} \over {dx}} + 2xy - x + 1 = 0\,\,\,$$ given that at $$x=1,$$ $$y=0$$ is

The number of boundary conditions required to solve the differential equation $$\,\,{{{\partial ^2}\phi } \over {\partial {x^2}}} + {{{\partial ^2}\phi } \over {\partial {y^2}}} = 0\,\,$$ is

If $$c$$ is a constant, then the solution of $${{dy} \over {dx}} = 1 + {y^2}$$ is

For the differential equation $$f\left( {x,y} \right){{dy} \over {dx}} + g\left( {x,y} \right) = 0\,\,$$ to be exact is