<p>The order and degree of the differential equation \(\left(1+3\dfrac{dy}{dx}\right)^{2/3} = 4\dfrac{d^3y}{dx^3}\) are</p>
Step-by-Step Solution
Key Concept: Order is determined by the highest derivative present (regardless of powers), while degree is the power of the highest derivative AFTER clearing all fractional powers and radicals from the equation.
<p><strong>Step 1:</strong> Identify the <strong>order</strong>.</p><p>The highest derivative present is d³y/dx³ (third derivative), so <strong>order = 3</strong>.</p><p><strong>Step 2:</strong> Find the <strong>degree</strong> by eliminating fractional powers.</p><p>Given: (1 + 3dy/dx)^(2/3) = 4(d³y/dx³)</p><p>Raise both sides to the power 3 to eliminate the fractional exponent:</p><p>[(1 + 3dy/dx)^(2/3)]³ = [4(d³y/dx³)]³</p><p>(1 + 3dy/dx)² = 64(d³y/dx³)³</p><p><strong>Step 3:</strong> Identify the power of the highest derivative.</p><p>After clearing radicals, the highest derivative d³y/dx³ appears with power 3, so <strong>degree = 3</strong>.</p><p>∴ Answer: <strong>Order = 3, Degree = 3</strong> (Option C)</p>
Correct Answer: C