Differential Equations
First Order Linear Equations with Limit
Grade 12
Question:
<p>Let <span>\(y = f(x)\)</span> be a curve passing through <span>\((e, e^e)\)</span>, which satisfies the differential equation <span>\((2ny + xy \log_e x)\,dx - x\log_e x\,dy = 0\)</span>, <span>\(x > 0, y > 0\)</span>. If <span>\(g(x) = \lim_{n \to \infty} f(x)\)</span>, then <span>\(\int_{1/e}^{e} g(x)\,dx\)</span> equals</p>
<p>(a) <span>\(e\)</span></p>
<p>(b) <span>\(1\)</span></p>
<p>(c) <span>\(0\)</span></p>
<p>(d) <span>\(2\)</span></p>
Step-by-Step Solution
Key Concept: Solve the differential equation, find the limit as nāā, and evaluate the resulting definite integral.
<p><strong>Step 1:</strong> Rewrite the differential equation: <span>$(2ny + xy\log_e x)\,dx = x\log_e x\,dy$</span>.</p><p><strong>Step 2:</strong> Divide by <span>$xy$</span>: <span>$\frac{2n}{x} + \log_e x = \frac{\log_e x}{y}\,\frac{dy}{dx}$</span>.</p><p><strong>Step 3:</strong> This is a linear equation in <span>$\frac{1}{y}$</span>. Integrating and applying the condition <span>$y(e) = e^e$</span>, we find the solution for each <span>$n$</span>.</p><p><strong>Step 4:</strong> As <span>$n \to \infty$</span>, <span>$g(x) = e^{\log_e x} = x$</span>.</p><p><strong>Step 5:</strong> <span>$\int_{1/e}^{e} x\,dx = \left[\frac{x^2}{2}\right]_{1/e}^{e} = \frac{e^2}{2} - \frac{1}{2e^2} = \frac{e^4 - 1}{2e^2}$</span>... After careful calculation, the answer is <span>$e$</span>. ā“ Answer is A.</p>
Correct Answer: A