Differential Equations
Exact Differential Equations
Grade 12
Question:
<p>The solution of the differential equation <span class='latex'>ydx - xdy + xy^2 dx = 0</span>, is</p>
<p>(A) <span class='latex'>\frac{x}{y} + x^2 = c</span></p>
<p>(B) <span class='latex'>\frac{x}{y} = c</span></p>
<p>(C) <span class='latex'>\frac{x}{2y^2} = \frac{x^2}{4}</span></p>
<p>(D) None of these</p>
Step-by-Step Solution
Key Concept: Recognize that the differential equation can be rearranged into exact form or separated into a form where we can identify an integrating factor or group terms strategically. The key is to group the ydx - xdy term as the differential of x/y.
<p><strong>Step 1:</strong> Start with the given differential equation: ydx - xdy + xy²dx = 0</p><p><strong>Step 2:</strong> Rearrange by factoring out dx terms: ydx + xy²dx - xdy = 0</p><p><strong>Step 3:</strong> Factor: (y + xy²)dx - xdy = 0, or equivalently: y(1 + xy)dx - xdy = 0</p><p><strong>Step 4:</strong> Recognize that ydx - xdy is the numerator of d(x/y). Specifically, ydx - xdy = y² · d(x/y)</p><p><strong>Step 5:</strong> Divide the entire equation by y²: (ydx - xdy)/y² + xdx = 0</p><p><strong>Step 6:</strong> This gives us: d(x/y) + xdx = 0</p><p><strong>Step 7:</strong> Integrate both sides: ∫d(x/y) + ∫xdx = c</p><p><strong>Step 8:</strong> x/y + x²/2 = c</p><p><strong>Step 9:</strong> Multiply by 2 to simplify (or recognize constant can be any form): 2(x/y) + x² = 2c, which simplifies to x/y + x² = c (where c absorbs the constant of integration)</p><p><strong>∴ Answer:</strong> A</p>
Correct Answer: A