Question

If $\dfrac{{dy}}{{dx}} = y + 3$ and $y(0) = 2$ then $y(\ln (2))$ is equal to:
A) $7$
B) $5$
C) $13$
D) $ - 2$

Answer 1

Hint: The given equation is a simple linear differential equation. So we just have to arrange terms of similar variables on one side and then simply integrate to get the function of $y$ in terms of $x$. Then we will use this relation to find desired value.

Complete step-by-step answer:
Given, $\dfrac{{dy}}{{dx}} = y + 3$,
After some rearrangements we get,
$\dfrac{{dy}}{{y + 3}} = dx$
Now, Integrate both sides,
\[\int {\dfrac{{dy}}{{y + 3}}} = \int {dx} \]
Since, $\int {\dfrac{{dx}}{x} = \ln (x} ) + c$ and $\int {dx = x + c} $
So we get,
\[\ln (y + 3) + {c_1} = x + {c_2}\]
On further solving we get,
$\ln (y + 3) = x + {c_2} - {c_1}$
This is equivalent to
$\ln (y + 3) = x + c$ --------(1)
Now given, $y(0) = 2$ So putting $x = 0$ and $y = 2$ in above equation we get,
\[\ln (2 + 3) = 0 + c\]
We get $c$ as,
$c = \ln (5)$
Using above value of $c$ in equation $1$ we get final equation as,
\[\ln (y + 3) = x + \ln (5)\]
On simplifying, we get,
$\ln (y + 3) - \ln (5) = x$
This simplifies to
\[\ln (\dfrac{{y + 3}}{5}) = x\] -------(2)
Because $\ln a - \ln b = \ln \dfrac{a}{b}$
Now, we want to find value of $y$ when $x = \ln (2)$
So, using equation $2$ we get,
$\ln \left( {\dfrac{{y + 3}}{5}} \right) = \ln (2)$
On comparing we get,
$\dfrac{{y + 3}}{5} = 2$
On simplification we get,
$y + 3 = 10$
So we get our answer as,
$y = 7$

Hence, the correct option is (A).

Note: We simplified two constant values in our solutions into a single constant ( ${c_2} - {c_1} = c$ ) because two constants, on subtraction, again gives a constant value which was equivalent to $\ln (5)$ . Otherwise, we would not be able to find two constant values separately.