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The Problem of Areas

If dydx=xy+...

Question

If $\frac{d y}{d x} = x y + 2 x + 3 y + 6$ , then find the value of $y (- 1) - e^{2} y (- 3)$ .

A

e^{2} - 1

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B

e^{2} + 1

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C

2 (e^{2} - 1)

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D

- 2 (e^{2} - 1)

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Solution

The correct option is C $2 (e^{2} - 1)$
$\frac{d y}{d x} = (x + 3) (y + 2)$ , If $X = x + 3, Y = y + 2$
$\frac{d Y}{Y} = X d X$
$\Rightarrow Y = A . e^{X^{2} / 2}, y = - 2 + A e^{(x + 3)^{2} / 2}$
$y (- 1) = - 2 + A e^{2}, y (- 3) = - 2 + A$
$y (- 1) - e^{2} y (- 3) = - 2 + 2 e^{2} = 2 (e^{2} - 1)$

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