Question

Answer 1

$\rm \longrightarrow \dfrac{dy}{dx} = 3y \\ \\ \\ \rm \longrightarrow \dfrac{dy}{y} = 3 \: dx\\ \\ \\ \rm \longrightarrow \int \dfrac{dy}{y} = \int 3 \: dx\\ \\ \\ \rm \longrightarrow \: log_{e}(y) + c_{1}= 3x + c_{2}\\ \\ \\ \rm \longrightarrow \: log_{e}(y) + c_{1} - c_{2}= 3x \\ \\ \\ \rm \longrightarrow \: log_{e}(y) + c= 3x ...(1)$

Now, to find the value of c put x = 0 and y = 2.

$\rm \longrightarrow \: log_{e}(2) + c= 0 \\ \\ \\ \rm \longrightarrow \: c= - log_{e}(2)$

Now, put the value of c in equation (1).

$\rm \longrightarrow \: log_{e}(y) - log_{e}(2)= 3x \\ \\ \rm \longrightarrow \: log_{e}(y) = 3x + log_{e}(2)\\ \\ \rm \longrightarrow \: y = {e}^{(3x + log_{e}(2))} \\ \\ \rm \longrightarrow \: y = {e}^{(3x )} \times {e}^{( log_{e}(2))} \\ \\ \rm \longrightarrow \: y = {e}^{(3x )} \times {2}^{( log_{e}(e))} \\ \\ \rm \longrightarrow \: y = {e}^{(3x )} \times {2}\\ \\ \rm \longrightarrow \: y = 2{e}^{(3x )}$

Therefore, option (b) is correct.