Question

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

• A. $7$
• B. $5$
• C. $13$
• D. $-2$

Answer 1

The correct option is A $7$

$\frac{dy}{dx}=y+3$ and y(0) = 2

then y (ln 2) = (?)

$\therefore \frac{dy}{dx}=y+3$

$\therefore \frac{1}{y+3}dy=dx$

Integrating both side

$\int \frac{1}{y+3}dy=\int dx$

$\therefore$ ln(y + 3) = x + c

$\to$ Initial condition y(0)=2

$\therefore ln(2+3)=0+c$

$\therefore c=ln5$

$\to$ ln(y + 3) = x + ln5

$\therefore$ ln(y + 3) - ln5 = x

$\therefore ln\left(\frac{y+3}{5}\right)=x$

$\therefore \frac{y+3}{5}=e^x$

$\therefore y=5e^x-3$

for y (ln2)

$y=5e^{ln2}-3$

$=5\times 2-3$

= 10 - 3

= 7