<p>The order and degree of the differential equation \(\left(1 + 3\dfrac{dy}{dx}\right)^{2/3} = 4\left(\dfrac{d^3y}{dx^3}\right)\) are respectively:</p>
Step-by-Step Solution
Key Concept: Order is determined by the highest derivative present (regardless of any powers), while degree is the power of the highest derivative after the equation is made polynomial in all derivatives. You must eliminate fractional powers by raising to appropriate powers.
<p><strong>Step 1: Identify the highest derivative.</strong></p><p>The equation contains d³y/dx³ (third derivative) and dy/dx (first derivative). The highest derivative present is d³y/dx³, so <strong>Order = 3</strong>.</p><p><strong>Step 2: Clear fractional exponents to find degree.</strong></p><p>The original equation is: (1 + 3dy/dx)^(2/3) = 4(d³y/dx³)</p><p>Raise both sides to the power of 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: Determine degree.</strong></p><p>After clearing fractional powers, the highest derivative d³y/dx³ appears with power 3. Therefore <strong>Degree = 3</strong>.</p><p><strong>Order = 3, Degree = 3</strong></p><p>∴ Answer: C</p>
Correct Answer: C