Question:
The real value of \( m \) for which the substitution \( y = u^m \) will transform the differential equation \( 2x^4 y \frac{dy}{dx} + y^4 = 4x^6 \) into a homogeneous equation is
The real value of \( m \) for which the substitution \( y = u^m \) will transform the differential equation \( 2x^4 y \frac{dy}{dx} + y^4 = 4x^6 \) into a homogeneous equation is
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In substitution problems, match exponents to make all terms of the same degree.
Updated On: Apr 23, 2026
- \( m = 0 \)
- \( m = 1 \)
- \( m = \frac{3}{2} \)
- \( m = \frac{2}{3} \)
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The Correct Option is C
Solution and Explanation
Concept:
To convert a differential equation into homogeneous form, all terms must have the same degree after substitution.
Step 1: Substitute \( y = u^m \). \[ \frac{dy}{dx} = m u^{m-1} \frac{du}{dx} \] Substitute: \[ 2x^4 (u^m)(m u^{m-1} \frac{du}{dx}) + u^{4m} = 4x^6 \] \[ 2m x^4 u^{2m-1} \frac{du}{dx} + u^{4m} = 4x^6 \]
Step 2: Match powers for homogeneity: \[ 4m = 6 \Rightarrow m = \frac{3}{2} \] Check: \[ 2m - 1 = 3 - 1 = 2 \quad (\text{consistent}) \] Final Answer: \[ m = \frac{3}{2} \]
Step 1: Substitute \( y = u^m \). \[ \frac{dy}{dx} = m u^{m-1} \frac{du}{dx} \] Substitute: \[ 2x^4 (u^m)(m u^{m-1} \frac{du}{dx}) + u^{4m} = 4x^6 \] \[ 2m x^4 u^{2m-1} \frac{du}{dx} + u^{4m} = 4x^6 \]
Step 2: Match powers for homogeneity: \[ 4m = 6 \Rightarrow m = \frac{3}{2} \] Check: \[ 2m - 1 = 3 - 1 = 2 \quad (\text{consistent}) \] Final Answer: \[ m = \frac{3}{2} \]
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