The emf \( \varepsilon \) induced in the circuit is given by Faraday’s law:
\[ \varepsilon = -\frac{d\Phi}{dt}. \]Calculate \( \frac{d\Phi}{dt} \):
\[ \frac{d\Phi}{dt} = 10t - 36. \]At \( t = 2 \, \text{s} \):
\[ \varepsilon = -(10 \cdot 2 - 36) = -(-16) = 16 \, \text{V}. \]The induced current \( i \) in the circuit is:
\[ i = \frac{\varepsilon}{R} = \frac{16}{8} = 2 \, \text{A}. \]Thus, the induced current at \( t = 2 \, \text{s} \) is:
\[ 2 \, \text{A}. \]If \[ \frac{dy}{dx} + 2y \sec^2 x = 2 \sec^2 x + 3 \tan x \cdot \sec^2 x \] and
and \( f(0) = \frac{5}{4} \), then the value of \[ 12 \left( y \left( \frac{\pi}{4} \right) - \frac{1}{e^2} \right) \] equals to: