You visited us 1 times! Enjoying our articles? Unlock Full Access!

Property 1

The real valu...

Question

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

m = 0

No worries! We‘ve got your back. Try BYJU‘S free classes today!

m = 1

No worries! We‘ve got your back. Try BYJU‘S free classes today!

m = \frac{3}{2}

Right on! Give the BNAT exam to get a 100% scholarship for BYJUS courses

m = \frac{2}{3}

No worries! We‘ve got your back. Try BYJU‘S free classes today!

Open in App

Solution

The correct option is B $m = \frac{3}{2}$
Given, $y = u^{m}$
$\Rightarrow \frac{d y}{d x} = m \cdot u^{m - 1} \cdot \frac{d u}{d x}$
Hence, $2 x^{4} \cdot m u^{m - 1} u^{m} \cdot \frac{d u}{d x} + u^{4 m} + 4 x^{6}$
$\Rightarrow \frac{d u}{d x} = \frac{4 x^{6} - u^{4 m}}{2 m x^{4} \cdot u^{2 m - 1}}$
Therefore, $4 m = 6$
$\Rightarrow m = \frac{3}{2}$

Suggest Corrections

Similar questions

y = u^{n}

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

n

y = u^{n}

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

y = z^{α}

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

I

II

\begin{matrix} C o l u m n I & C o l u m n I I a. If the function y = e^{4 x} + 2 e^{- x} is a solution of the differential equation \frac{\frac{d^{3} y}{d x^{3}} - 13 \frac{d y}{d x}}{y} = K, then the value of \frac{K}{3} is & p. 3 b. Number of straight lines which satisfy the differential equation \frac{d y}{d x} + x {(\frac{d y}{d x})}^{2} - y = 0 is & q. 4 c. If real value of m for which the substitution, y = u^{m} will transform the differential equat- ion, 2 x^{4} y \frac{d y}{d x} + y^{4} = 4 x^{6} into a homogeneous equation, then the value of 2 m is & r. 2 d. If the solution of differential equation x^{2} \frac{d^{2} y}{d x^{2}} + 2 x \frac{d y}{d x} = 12 y is y = A x^{m} + B x^{- n}, then | m + n | is & s. 7 \end{matrix}

a

m

(a^{2} - 1) m^{2} - (2 a - 3) m + a = 0

y -

Join BYJU'S Learning Program