Let y = y ( x ) be the solution of the differential equation ( 3 y 2 - 5 x 2 ) y d x + 2 x ( x 2 - y 2 ) d y = 0 such that y ( 1 ) = 1 . Then | ( y ( 2 ) ) 3 - 12 y ( 2 ) | is equal to [2023]

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Solution

Q.
Let $y = y (x)$ be the solution of the differential equation $(3 y^{2} - 5 x^{2}) y d x + 2 x (x^{2} - y^{2}) d y = 0$ such that $y (1) = 1$ . Then $| (y {(2))}^{3} - 12 y (2) |$ is equal to [2023]

1 $64$

2 $32 \sqrt{2}$

3 $32$

4 $16 \sqrt{2}$

Ans.
(2)

$Given, (3 y^{2} - 5 x^{2}) y d x + 2 x (x^{2} - y^{2}) d y = 0$

$\Rightarrow \frac{d y}{d x} = \frac{y (5 x^{2} - 3 y^{2})}{2 x (x^{2} - y^{2})}$

It is a homogeneous differential equation.

Put $y = m x \Rightarrow$ $\frac{d y}{d x} = m + x \frac{d m}{d x}$

$\Rightarrow m + x \frac{d m}{d x} = \frac{m (5 - 3 m^{2})}{2 (1 - m^{2})}$ $\Rightarrow \frac{x d m}{d x} = \frac{m (5 - 3 m^{2})}{2 (1 - m^{2})} - m$

$\Rightarrow x \frac{d m}{d x} = \frac{(5 - 3 m^{2}) m - 2 m (1 - m^{2})}{2 (1 - m^{2})}$

$\Rightarrow \frac{d x}{x} = \frac{2 (m^{2} - 1)}{m (m^{2} - 3)} d m$ $\Rightarrow \frac{d x}{x} = [\frac{2}{m} - \frac{\frac{4}{3}}{m} + \frac{\frac{4 m}{3}}{m^{2} - 3}] d m$

Integrating both sides $\int \frac{d x}{x} = \int \frac{(\frac{2}{3})}{m} d m + \int \frac{2}{3} (\frac{2 m}{m^{2} - 3}) d m$

$\Rightarrow \ln$ $| x |$ $= \frac{2}{3} \ln$ $| m |$ $+ \frac{2}{3} \ln$ $| m^{2} - 3 |$ $+ C$

$\Rightarrow \ln$ $| x |$ $= \frac{2}{3} \ln$ $| \frac{y}{x} |$ $+ \frac{2}{3} \ln$ $| {(\frac{y}{x})}^{2} - 3 |$ $+ C$

Put $(x = 1, y = 1)$ , $\ln (1) = \frac{2}{3} \ln (1) + \frac{2}{3} \ln$ $| (1 - 3) |$ $+ C$

$C = - \frac{2}{3} \ln (2) \Rightarrow \ln$ $| x |$ $= \frac{2}{3} \ln$ $| \frac{y}{x} |$ $+ \frac{2}{3} \ln$ $| {(\frac{y}{x})}^{2} - 3 |$ $- \frac{2}{3} \ln (2)$

$\Rightarrow (\frac{y}{x}) [{(\frac{y}{x})}^{2} - 3] = 2 (x^{3 / 2})$

Put $x = 2$ to get $y (2)$ , $\Rightarrow y (y^{2} - 12) = 4 \times 2 \times 2 \times 2 \sqrt{2}$

$\Rightarrow y^{3} - 12 y = 32 \sqrt{2} \Rightarrow$ $| y^{3} (2) - 12 y (2) |$ $= 32 \sqrt{2}$

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