Differential Equations
Formation of Differential Equations
Grade 12
Question:
<p>The degree and order of the differential equation on the family of all parabolas whose axis is <em>x</em>-axis are, respectively,</p>
<p>2, 1</p>
<p>1, 2</p>
<p>3, 2</p>
<p>2, 3</p>
Step-by-Step Solution
Key Concept: The family of parabolas with x-axis as axis has equation y² = 4ax (containing parameter 'a'). Eliminate the parameter by successive differentiation to find the actual differential equation, then identify its order (highest derivative) and degree (power of highest derivative).
<p><strong>Step 1:</strong> Family of parabolas with x-axis as axis: <strong>y² = 4ax</strong> (one parameter 'a')</p><p><strong>Step 2:</strong> Differentiate both sides with respect to x: <br/>2y(dy/dx) = 4a<br/>Therefore: <strong>a = y(dy/dx)/2</strong></p><p><strong>Step 3:</strong> Substitute back into y² = 4ax:<br/>y² = 4 · [y(dy/dx)/2] · x<br/>y² = 2xy(dy/dx)<br/>If y ≠ 0: <strong>y = 2x(dy/dx)</strong></p><p><strong>Step 4:</strong> This can be written as: <strong>2x(dy/dx) - y = 0</strong></p><p><strong>Step 5:</strong> Identify order and degree:<br/>• <strong>Order</strong> = highest derivative present = 1 (only dy/dx)<br/>• <strong>Degree</strong> = power of highest derivative = 1 (dy/dx appears to power 1)</p><p>∴ Answer: <strong>Order = 1, Degree = 1</strong> (or <strong>1, 1</strong>)</p>
Correct Answer: B