Question

The real value for $n$ for which the substitution $y=u^n$ will transform the differential equation $2x^{4}y\frac{dy}{dx}+y^4=4x^4$ into a homogeneous equation is

• A. $\frac{1}{2}$
• B. $1$
• C. $\frac{3}{2}$
• D. $2$

Answer 1

The correct option is A $\frac{1}{2}$

Given differential equation is $2x^{4}y\frac{dy}{dx}+y^4=4x^4$,

Substituting $y=u^n$

$⟹\frac{dy}{dx}=n{u}^{n-1}\frac{du}{dx}$ gives,

$⟹2x^4{u}^n\left(n{u}^{n-1}\frac{du}{dx}\right)=4x^4-u^{4n}⟹\frac{du}{dx}=\frac{2}{n{u}^{2n-1}}-\frac{u^{2n+1}}{2n{x}^4}$,

Given this equation is homogenous,

By inspection,if $n=\frac{1}{2}$,

The equation becomes $\frac{du}{dx}+\frac{u^2}{x^4}=4$, which is homogenous.