Question:
The order and degree of the differential equation : \( \frac{d}{dx}(y')^3 + (y')^3 = 1 \) respectively are where \( y' = \frac{dy}{dx} \)
The order and degree of the differential equation : \( \frac{d}{dx}(y')^3 + (y')^3 = 1 \) respectively are where \( y' = \frac{dy}{dx} \)
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Always perform indicated operations like \( \frac{d}{dx} \) before determining the order and degree.
Ensure all derivatives are free from radicals or fractional exponents to correctly identify the degree.
Ensure all derivatives are free from radicals or fractional exponents to correctly identify the degree.
Updated On: Sep 10, 2026
- \( 1, 3 \)
- \( 2, 1 \)
- \( 3, 1 \)
- \( 3, 2 \)
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The Correct Option is B
Solution and Explanation
Concept:
• Order: The order of a differential equation is the order of the highest derivative present in the equation.
• Degree: The degree is the power of the highest order derivative, provided the equation is a polynomial in its derivatives.
Step 1: Expand the given differential equation
The given equation is: \[ \frac{d}{dx}(y')^3 + (y')^3 = 1 \] We need to perform the differentiation in the first term with respect to \( x \). Using the chain rule: \[ \frac{d}{dx} \left( \frac{dy}{dx} \right)^3 = 3 \left( \frac{dy}{dx} \right)^2 \cdot \frac{d}{dx} \left( \frac{dy}{dx} \right) \] \[ = 3(y')^2 \cdot y'' \]
Step 2: Rewrite the full equation
Substituting the derivative back into the original equation: \[ 3(y')^2 y'' + (y')^3 = 1 \] Or in standard notation: \[ 3 \left( \frac{dy}{dx} \right)^2 \left( \frac{d^2y}{dx^2} \right) + \left( \frac{dy}{dx} \right)^3 = 1 \]
Step 3: Identify the order and degree
The highest derivative present in the equation is the second derivative, \( y'' \) or \( \frac{d^2y}{dx^2} \). Therefore, the order is 2. The highest order derivative term is \( 3(y')^2 y'' \). The exponent (power) of this term \( (y'') \) is 1. Therefore, the degree is 1.
• Order: The order of a differential equation is the order of the highest derivative present in the equation.
• Degree: The degree is the power of the highest order derivative, provided the equation is a polynomial in its derivatives.
Step 1: Expand the given differential equation
The given equation is: \[ \frac{d}{dx}(y')^3 + (y')^3 = 1 \] We need to perform the differentiation in the first term with respect to \( x \). Using the chain rule: \[ \frac{d}{dx} \left( \frac{dy}{dx} \right)^3 = 3 \left( \frac{dy}{dx} \right)^2 \cdot \frac{d}{dx} \left( \frac{dy}{dx} \right) \] \[ = 3(y')^2 \cdot y'' \]
Step 2: Rewrite the full equation
Substituting the derivative back into the original equation: \[ 3(y')^2 y'' + (y')^3 = 1 \] Or in standard notation: \[ 3 \left( \frac{dy}{dx} \right)^2 \left( \frac{d^2y}{dx^2} \right) + \left( \frac{dy}{dx} \right)^3 = 1 \]
Step 3: Identify the order and degree
The highest derivative present in the equation is the second derivative, \( y'' \) or \( \frac{d^2y}{dx^2} \). Therefore, the order is 2. The highest order derivative term is \( 3(y')^2 y'' \). The exponent (power) of this term \( (y'') \) is 1. Therefore, the degree is 1.
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