Differential Equations
Formation of Differential Equations
Grade 12
Question:
<p>If the solution of the differential equation is \(Ax^2 + By^2 = 1\), then the order and degree of the resulting differential equation (after eliminating \(A\) and \(B\)) are:</p>
<p>Order = 2, Degree = 1</p>
<p>Order = 1, Degree = 2</p>
<p>Order = 2, Degree = 2</p>
<p>Order = 1, Degree = 1</p>
Step-by-Step Solution
Key Concept: Differentiate the solution curve twice to eliminate the two arbitrary constants A and B, then identify the highest derivative order and its power in the resulting equation.
<p><strong>Step 1:</strong> Start with the solution curve: <em>Ax</em>² + <em>By</em>² = 1</p><p><strong>Step 2:</strong> Differentiate once with respect to <em>x</em>:<br/>2<em>Ax</em> + 2<em>By</em>·<em>dy/dx</em> = 0<br/>⟹ <em>Ax</em> + <em>By</em>·<em>dy/dx</em> = 0 ... (i)</p><p><strong>Step 3:</strong> Differentiate equation (i) again:<br/><em>A</em> + <em>B</em>·<em>d</em>²<em>y/dx</em>² + <em>B</em>·(<em>dy/dx</em>)² = 0 ... (ii)</p><p><strong>Step 4:</strong> From original equation: <em>A</em> = (1 − <em>By</em>²)/<em>x</em>²<br/>From equation (i): <em>A</em> = −<em>By</em>·(<em>dy/dx</em)/<em>x</em><br/>Equating: (1 − <em>By</em>²)/<em>x</em>² = −<em>By</em>·(<em>dy/dx</em)/<em>x</em></p><p><strong>Step 5:</strong> Solve for <em>B</em> and substitute back into equation (ii) to eliminate both <em>A</em> and <em>B</em>. The resulting equation will contain <em>d</em>²<em>y/dx</em>² as the highest derivative.</p><p><strong>Step 6:</strong> The resulting differential equation has:<br/>• <strong>Order</strong> = 2 (highest derivative is <em>d</em>²<em>y/dx</em>²)<br/>• <strong>Degree</strong> = 1 (power of <em>d</em>²<em>y/dx</em>² is 1)</p><p>∴ Answer: <strong>Order = 2, Degree = 1</strong></p>
Correct Answer: A