If [dy dx = y + 3 0,,] and y(0)=2, then [y( 2 )] is equal to :

Question

If $${{dy} \over {dx}} = y + 3 > 0\,\,$$ and $y(0)=2,$ then $$y\left( {\ln 2} \right)$$ is equal to :

Accepted Answer

$${{dy} \over {dx}} = y + 3 \Rightarrow \int {{{dy} \over {y + 3}}}  = \int {dx} $$

$$ \Rightarrow \ell n\left| {y + 3} \right| = x + c$$

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

$$\ell n\left| {y + 3} \right| = \ell n2 + \ell n5$$

$$ \Rightarrow \ln \left| {y + 3} \right| = \ell n\,10$$

$$\therefore$$ $$\,\,\,\,\,$$ $$y + 3 =  \pm 10 \Rightarrow y = 7, - 13$$

Solutions · 1