Differential Equations
First Order Linear Equations
Grade 12
Question:
<p>Solution of differential equation <span style='font-style:italic;'>x</span> cos <span style='font-style:italic;'>x</span> \(\left(\frac{dy}{dx}\right)\) + <span style='font-style:italic;'>y</span>(<span style='font-style:italic;'>x</span> sin <span style='font-style:italic;'>x</span> + cos <span style='font-style:italic;'>x</span>) = 1 is:</p>
<p>(a) <span style='font-style:italic;'>xy</span> = sin <span style='font-style:italic;'>x</span> + <span style='font-style:italic;'>c</span> cos <span style='font-style:italic;'>x</span></p>
<p>(b) <span style='font-style:italic;'>xy</span> sec <span style='font-style:italic;'>x</span> = tan <span style='font-style:italic;'>x</span> + <span style='font-style:italic;'>c</span></p>
<p>(c) <span style='font-style:italic;'>xy</span> + sin <span style='font-style:italic;'>x</span> + <span style='font-style:italic;'>c</span> cos <span style='font-style:italic;'>x</span> = 0</p>
<p>(d) None of these</p>
Step-by-Step Solution
Key Concept: Recognize this as a linear differential equation in standard form dy/dx + P(x)y = Q(x). Divide by the coefficient of dy/dx, then find an integrating factor to convert the left side into a perfect derivative.
<p><strong>Step 1:</strong> Rewrite the differential equation in standard form.</p><p>Given: x cos x (dy/dx) + y(x sin x + cos x) = 1</p><p>Divide throughout by x cos x:</p><p>dy/dx + y(x sin x + cos x)/(x cos x) = 1/(x cos x)</p><p>dy/dx + y(sin x/cos x + 1/x) = 1/(x cos x)</p><p>dy/dx + y(tan x + 1/x) = sec x/x</p><p><strong>Step 2:</strong> Identify P(x) and Q(x).</p><p>Here, P(x) = tan x + 1/x and Q(x) = sec x/x</p><p><strong>Step 3:</strong> Find the integrating factor μ(x).</p><p>μ(x) = e^(∫P(x)dx) = e^(∫(tan x + 1/x)dx)</p><p>= e^(ln|sec x| + ln|x|) = e^(ln|x sec x|) = x sec x</p><p><strong>Step 4:</strong> Multiply the equation by integrating factor x sec x.</p><p>x sec x · dy/dx + x sec x · y(tan x + 1/x) = x sec x · sec x/x</p><p>x sec x · dy/dx + y sec x(x tan x + sec x) = sec² x</p><p>d/dx(xy sec x) = sec² x</p><p><strong>Step 5:</strong> Integrate both sides.</p><p>xy sec x = ∫sec² x dx = tan x + c</p><p><strong>Step 6:</strong> Verify this matches option (b).</p><p>xy sec x = tan x + c ✓ [This is option (b)]</p><p><strong>Step 7:</strong> Verify option (a) is equivalent.</p><p>From xy sec x = tan x + c, multiply both sides by cos x:</p><p>xy sec x · cos x = (tan x + c) cos x</p><p>xy = sin x + c cos x ✓ [This is option (a)]</p><p><strong>∴ Answer:</strong> a, b</p>
Correct Answer: a, b