Differential Equations Question - None

Question

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) respectively, 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).

Accepted Answer

Step 1: Form the difference equation

Let w(x) = u(x) - v(x). Subtract the equations:

dw/dx + p(x)w = [f(x) - g(x)]

Step 2: Analyze the sign of dw/dx

Since f(x) > g(x) for all x > x₁, we have:

dw/dx = f(x) - g(x) - p(x)w(x)

More directly: dw/dx = [f(x) - g(x)] - p(x)w

Step 3: Use the integrating factor method

Multiply by integrating factor μ(x) = e^(∫p(x)dx):

d/dx[e^(∫p(x)dx) · w(x)] = e^(∫p(x)dx) · [f(x) - g(x)]

Step 4: Integrate from x₁ to x

For x > x₁:

e^(∫p(x)dx) · w(x) = e^(∫p(x₁)dx) · w(x₁) + ∫_{x₁}^{x} e^(∫p(t)dx) · [f(t) - g(t)] dt

Step 5: Conclude the proof

Since w(x₁) = u(x₁) - v(x₁) > 0 and the integral ∫_{x₁}^{x} e^(∫p(t)dx) · [f(t) - g(t)] dt > 0 for x > x₁ (both terms positive), we have w(x) > 0 for all x > x₁.

Therefore u(x) > v(x) for x > x₁, so the curves y = u(x) and y = v(x) do not intersect for any point where x > x₁. ∴ Proof complete.

Question:

Step-by-Step Solution

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