JEE MAIN - Mathematics (2011 - No. 9)
If $${{dy} \over {dx}} = y + 3 > 0\,\,$$ and $$y(0)=2,$$ then $$y\left( {\ln 2} \right)$$ is equal to :
$$5$$
$$13$$
$$-2$$
$$7$$
Explanation
$${{dy} \over {dx}} = y + 3 \Rightarrow \int {{{dy} \over {y + 3}}} = \int {dx} $$
$$ \Rightarrow \ell n\left| {y + 3} \right| = x + c$$
Since $$y\left( 0 \right) = 2,\,\,\,$$ $$\,\,\,\,\,\,\,$$ $$\therefore$$ $$\,\,\,\,\,\,\,$$ $$\ell n5 = c$$
$$ \Rightarrow \ell n\left| {y + 3 = x + \ell n5} \right|$$
When $$x = \ell n2,$$ then
$$\ell n\left| {y + 3} \right| = \ell n2 + \ell n5$$
$$ \Rightarrow \ln \left| {y + 3} \right| = \ell n\,10$$
$$\therefore$$ $$\,\,\,\,\,$$ $$y + 3 = \pm 10 \Rightarrow y = 7, - 13$$
$$ \Rightarrow \ell n\left| {y + 3} \right| = x + c$$
Since $$y\left( 0 \right) = 2,\,\,\,$$ $$\,\,\,\,\,\,\,$$ $$\therefore$$ $$\,\,\,\,\,\,\,$$ $$\ell n5 = c$$
$$ \Rightarrow \ell n\left| {y + 3 = x + \ell n5} \right|$$
When $$x = \ell n2,$$ then
$$\ell n\left| {y + 3} \right| = \ell n2 + \ell n5$$
$$ \Rightarrow \ln \left| {y + 3} \right| = \ell n\,10$$
$$\therefore$$ $$\,\,\,\,\,$$ $$y + 3 = \pm 10 \Rightarrow y = 7, - 13$$
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