Differential Equations
Solution of Differential Equations
Grade 12
Question:
<p>The solution of the differential equation \(y\,dx + (x + x^2 y)\,dy = 0\) is:</p>
<p>(1) \(\ln y = c\)</p>
<p>(2) \(-\dfrac{1}{xy} + \ln y = c\)</p>
<p>(3) \(\dfrac{1}{xy} + \ln y = c\)</p>
<p>(4) \(-\dfrac{1}{xy} - \ln y = c\)</p>
Step-by-Step Solution
Key Concept: Recognize this as an exact differential equation by rearranging as M dx + N dy = 0, then verify ∂M/∂y = ∂N/∂x to confirm exactness before solving.
<p><strong>Step 1:</strong> Rewrite the equation in standard form M dx + N dy = 0</p><p>y dx + (x + x²y) dy = 0</p><p>Here: M = y, N = x + x²y</p><p><strong>Step 2:</strong> Check exactness: ∂M/∂y = 1 and ∂N/∂x = 1 + 2xy</p><p>Not exact as written. Rearrange: y dx + x dy + x²y dy = 0</p><p><strong>Step 3:</strong> Notice that y dx + x dy = d(xy)</p><p>So: d(xy) + x²y dy = 0</p><p><strong>Step 4:</strong> Divide by y (y ≠ 0): d(xy)/y + x² dy = 0</p><p>This gives: d(xy) + x²y dy = 0</p><p><strong>Step 5:</strong> Recognize that d(xy) + x²y dy can be rewritten. Let u = xy:</p><p>du + u·x dy = 0 is complex. Instead, observe:</p><p>d(xy) + x² d(y) can be regrouped as d(xy + x³y/3) through integration</p><p>Or more directly: xy + (x³y)/3 = C</p><p>Multiply by 3: <strong>3xy + x³y = 3C</strong> or <strong>xy(3 + x²) = C</strong></p><p>∴ Answer: <strong>xy(3 + x²) = C</strong> or equivalent form</p>
Correct Answer: B