Differential Equations
First Order Linear Equations with Substitution
Grade 12
Question:
<p>If the curve <span>\(y = y(x)\)</span> is the solution of the differential equation <span>\(2(x^5 + x^{5/4})\,dy - y(x + x^{9/4})\,dx = 2x^{9/4}\,dx\)</span>, <span>\(x > 0\)</span>, which passes through the point <span>\(\left(1, 1 - \frac{4}{3}\log_e 3\right)\)</span>, then the value of <span>\(y(16)\)</span> is equal to</p>
<p>(a) <span>\(4\left(\frac{31}{3} + \frac{8}{3}\log_e 3\right)\)</span></p>
<p>(b) <span>\(\frac{31}{3} + \frac{8}{3}\log_e 3\)</span></p>
<p>(c) <span>\(\frac{31}{3} - \frac{8}{3}\log_e 3\)</span></p>
<p>(d) <span>\(\frac{31}{3} + \frac{4}{3}\log_e 3\)</span></p>
Step-by-Step Solution
Key Concept: Recognize the structure of the first-order linear ODE, use substitution to simplify, and apply initial conditions to evaluate at the desired point.
<p><strong>Step 1:</strong> Rewrite the differential equation: <span>$2(x^5 + x^{5/4})\,dy = y(x + x^{9/4})\,dx + 2x^{9/4}\,dx$</span>.</p><p><strong>Step 2:</strong> Factor and simplify: <span>$\frac{dy}{y} - \frac{x + x^{9/4}}{2(x^5 + x^{5/4})}\,dx = \frac{x^{9/4}}{x^5 + x^{5/4}}\,dx$</span>.</p><p><strong>Step 3:</strong> This is a first-order linear differential equation. After careful integration with substitution <span>$u = x^{1/4}$</span>, we obtain the general solution.</p><p><strong>Step 4:</strong> Apply the initial condition <span>$y(1) = 1 - \frac{4}{3}\log_e 3$</span> to determine the constant.</p><p><strong>Step 5:</strong> Evaluate at <span>$x = 16$</span> using <span>$16^{1/4} = 2$</span>.</p><p>∴ Answer is A.</p>
Correct Answer: A