If , x(1) = 1, then 5x(2) is equal to __________. [2024]

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Solution

Q.
If $\frac{d x}{d y} = \frac{1 + x - y^{2}}{y}$ , x(1) = 1, then 5x(2) is equal to __________. [2024]

Ans.
(5)

We have, $\frac{d x}{d y} = \frac{1 + x - y^{2}}{y} \Rightarrow \frac{d x}{d y} - \frac{x}{y} = \frac{1 - y^{2}}{y}$

Now, $I . F . = e^{- \int \frac{1}{y} d y} = e^{- \log y} = \frac{1}{y}$

So, solution is given by,

$\frac{x}{y} = \int \frac{1 - y^{2}}{y^{2}} d y + C \Rightarrow \frac{x}{y} = \frac{- 1}{y} - y + C$

$\Rightarrow x = - 1 - y^{2} + C y$

$∵ x (1) = 1 \Rightarrow 1 = - 1 - 1 + C \Rightarrow C = 3$

$\Rightarrow x = - 1 - y^{2} + 3 y$

$∴$ 5x(2) = 5(–1 – 4 + 6) = 5.

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