Let u (x) and v(x) satisfy the differential equations du/dx + p(x) u = f(x) and dv/dx + p (x) v = g (x), where p(x), f(x) and g(x) are continuous functions. If u(x1) > v (x1) for some x1 and f(x) > g (x) for all x > x1, prove that any point (x, y) where x > x1 does not satisfy the equations y = u (x) and y = v (x).
Let w (x) = u (x) - v (x) ....(i)
and h (x) = f (x) - g (x)
On differentiating Eq. (i) w.r.t. x
Hence, there cannot exist a point (x, y ) such that x > x1 and y = u (x) and y = v (x).
