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General Solution of a Differential Equation

The real valu...

Question

The real value of n for which substitution $y = u^{n}$ will transform differential equation $2 x^{4} y \frac{d y}{d x} + y^{4} = 4 x^{6}$ into homogeneous equation

\frac{1}{2}

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\frac{3}{2}

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Solution

The correct option is C $\frac{3}{2}$
$2 x^{4} y \frac{d y}{d x} + y^{4} = 4 x^{6}$

$2 y \frac{d y}{d x} + \frac{y^{4}}{x^{4}} = 4 x^{2}$

$y = u^{n}$

$\frac{d y}{d x} = n x^{n - 1}$

$2 n u^{n - 1} u^{n} \frac{d u}{d x} + \frac{u^{4 x}}{x^{4}} = 4 x^{2}$

$2 x - 1 = 2$

$2 x = 3$

$x = \frac{3}{2}$

Suggest Corrections

Similar questions

n

y = u^{n}

2 x^{4} y \frac{d y}{d x} + y^{4} = 4 x^{4}

m

y = u^{m}

2 x^{4} \cdot y \frac{d y}{d x} + y^{4} = 4 x^{6}

y = z^{α}

(x^{2} y^{2} - 1) d y + 2 x y^{3} d x = 0

a_{1} \frac{d^{2} y}{d x^{2}} + a_{2} y \frac{d y}{d x} + a_{3} y = a_{4}

a_{1} \frac{d^{3} y}{d x^{3}} + a_{2} y = a_{3}

a_{1} \frac{d^{3} y}{d x^{2}} + a_{2} x \frac{d y}{d x} + a_{3} x^{2} y = 0

\frac{d x}{d y} = h (\frac{x}{y})

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