समीकरण z = ax2 + by2 से स्वेच्छ अचरों (arbitrary constants) 'a' और 'b' को हटाकर एक आंशिक अवकल समीकरण (partial differential equation) बनाइए, जहाँ p = ∂z∂x और q = ∂z∂y हैं।

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: