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

• A. \( m = 0 \)
• B. \( m = 1 \)
• C. \( m = \frac{3}{2} \)
• D. \( m = \frac{2}{3} \)

Hint:

In substitution problems, match exponents to make all terms of the same degree.

Answer 1

The Correct Option is C

To solve this problem, we need to transform the given differential equation into a homogeneous form using the substitution \( y = u^m \). Let's do this step-by-step: 

1. Given the differential equation:
2. Substitute \( y = u^m \). Then \( \frac{dy}{dx} = mu^{m-1} \frac{du}{dx} \).
3. Substitute these into the differential equation:
4. Simplify the expression:
5. Factor out the common term \(u^{2m}\):
6. The transformed equation should be homogeneous, which means the powers of \(x\) on both sides should be equal.
7. For the left side to be equal to the right side, the exponents of \(u\) must result in the same degree as the \(x^6\) term. Therefore:
8. Solve for \(m\):
9. Thus, the correct value of \( m \) that transforms the equation into a homogeneous equation is:

Hence, the correct answer is \( m = \frac{3}{2} \).