For a ∈ ℝ, let y1(x) and y2(x) be solutions of the differential equation: y'' + (ex2 + cos x)y = 0 such that y1(0) = 3, y1'(0) = -1, y2(0) = -5, y2'(0) = a. Suppose W(y1, y2)(x) denotes the Wronskian of y1 and y2. If W(y1, y2)12 = 4, then the value of a is:

The integrating factor for the linear differential equation (1 + x2)dydx + 2xy = cot(x) is:

The differential equation corresponding to the family of curves y2 = 4ax, where 'a' is an arbitrary constant, is:

The general solution of the differential equation y' = y, where 'a' is an arbitrary constant, is:

The particular solution of the differential equation dydx = 3x2 with initial condition y(1) = 4 is:

The solution of the differential equation x(1 + y2)dx + y(1 + x2)dy = 0 is:

The solution of the differential equation dydx = 2x, given that y(0) = 6, is:

The general solution of the differential equation dydx = e-y(ex + x2) is: