Differential Equations
Formation of Differential Equations
Grade 12
Question:
<p>The differential equation of all non-vertical lines in a plane is</p>
<p>\(\dfrac{d^2y}{dx^2} = 0\)</p>
<p>\(\dfrac{d^2x}{dy^2} = 0\)</p>
<p>\(\dfrac{dy}{dx} = 0\)</p>
<p>\(\dfrac{dx}{dy} = 0\)</p>
Step-by-Step Solution
Key Concept: A non-vertical line has constant slope m, so differentiation with respect to x eliminates the arbitrary constant and gives the differential equation directly without involving the y-intercept.
<p><strong>Step 1:</strong> A non-vertical line has the general form y = mx + c, where m and c are arbitrary constants.</p><p><strong>Step 2:</strong> To eliminate arbitrary constants, differentiate both sides with respect to x: dy/dx = m</p><p><strong>Step 3:</strong> Since m is constant, the second derivative is: d²y/dx² = 0</p><p><strong>Step 4:</strong> The differential equation d²y/dx² = 0 represents all non-vertical lines in the plane, as it contains no arbitrary constants and is satisfied by every non-vertical line.</p><p>∴ Answer: <strong>d²y/dx² = 0</strong> (or equivalently, dy/dx = constant)</p>
Correct Answer: A