Differential Equations
Exact Differential Equations
Grade 12
Question:
<p>Solution of the differential equation <span style='font-family: Arial, sans-serif;'>(x<sup>2</sup> − ay)dx − (ax − y<sup>2</sup>)dy = 0</span> is</p>
<p>(a) x<sup>3</sup> + y<sup>3</sup> − 3axy = 3C</p>
<p>(b) x<sup>3</sup> + y<sup>3</sup> + 3axy = C</p>
<p>(c) x<sup>3</sup> + y<sup>3</sup> − axy = C</p>
<p>(d) x<sup>3</sup> + y<sup>3</sup> − 3xy = 3C</p>
Step-by-Step Solution
Key Concept: An exact differential equation satisfies ∂M/∂y = ∂N/∂x. Integrate M with respect to x and non-x terms of N with respect to y, then sum to get the solution.
<p><strong>Step 1:</strong> Check if the equation is exact. Write as <i>M dx</i> + <i>N dy</i> = 0 where <i>M</i> = x<sup>2</sup> − ay and <i>N</i> = −(ax − y<sup>2</sup>).</p><p><strong>Step 2:</strong> Verify exactness: ∂<i>M</i>/∂<i>y</i> = −a and ∂<i>N</i>/∂<i>x</i> = −a, so ∂<i>M</i>/∂<i>y</i> = ∂<i>N</i>/∂<i>x</i>. The equation is exact.</p><p><strong>Step 3:</strong> Integrate <i>M</i> with respect to <i>x</i> treating <i>y</i> as constant: ∫(x<sup>2</sup> − ay)dx = x<sup>3</sup>/3 − axy.</p><p><strong>Step 4:</strong> From <i>N</i>, integrate terms not containing <i>x</i>: −∫(−y<sup>2</sup>)dy = y<sup>3</sup>/3.</p><p><strong>Step 5:</strong> Equate sum to constant: x<sup>3</sup>/3 − axy + y<sup>3</sup>/3 = C₁, which gives x<sup>3</sup> + y<sup>3</sup> + 3axy = C (after rearrangement and multiplication by 3).</p><p>∴ Answer is (b).</p>
Correct Answer: b