Question

The real value of m for which the substitution y = u^m will transform the differential equation 2x⁴y $\frac{dy}{dx}$ + y⁴ = 4x⁶ in to a homogeneous equation is

• A. m = 0
• B. m = 1
• C. m = 3/2
• D. m = 2/3

Answer 1

Answer: (C)

m = 3/2

Answer 2

y = u^m

dy/dx = mu^m–1 $\frac{du}{dx}$

The given differential equation becomes

2x⁴.u^m. mu^m–1 $\frac{du}{dx}$ + u^4m = 4x⁶

̃$\frac{du}{dx}$= $\frac{4x^{6} - u^{4m}}{2mx^{4}x^{2m - 1}}$

For homogeneous equation degree should be same in

numerator & denominator so,

6 = 4m = 4 + 2m – 1 ̃ m = 3/2