<p>If the general solution of a differential equation is \((y + c)^2 = cx\), where \(c\) is an arbitrary constant, then the order of the differential equation is _____.</p>
Step-by-Step Solution
Key Concept: The order of a differential equation equals the number of arbitrary constants in its general solution. Since the given general solution contains one arbitrary constant c, the differential equation must be first-order.
<p><strong>Step 1:</strong> Identify the general solution: $(y + c)^2 = cx$, where $c$ is an arbitrary constant.</p><p><strong>Step 2:</strong> Count the number of independent arbitrary constants in the general solution.</p><p>The general solution contains exactly <strong>one arbitrary constant</strong> ($c$).</p><p><strong>Step 3:</strong> Apply the fundamental principle: The order of a differential equation equals the number of arbitrary constants in its general solution.</p><p><strong>Step 4:</strong> Since there is 1 arbitrary constant, the order is 1.</p><p><strong>Verification:</strong> If we eliminate $c$ from $(y+c)^2 = cx$ by differentiating, we get a first-order DE: $2(y+c)\frac{dy}{dx} = c$, confirming the order is 1.</p><p>∴ <strong>Answer: 1</strong></p>
Correct Answer: 1