The switch is closed at $t=0$, find values of $i(t):

Solution:

$\because$ All initial conditions are zero $\therefore I_L(\overline{0})=I_0=0$ Amp ckt at $t=0^{+}$

Apply KVL to loop

$ \begin{gathered} \\ 100-10 i(t)-1 \cdot \frac{d i(t)}{d t}=0...(1) \\\\ \text { put } t=0^{+}: \frac{d i\left(0^{+}\right)}{d t}=0 \\\\ 100-10 i\left(0^{+}\right)-\frac{d i\left(0^{+}\right)}{d t}=100 \quad \text { A/s } \\ \end{gathered} \\ $

Differentiating $e q-n$ (1)

$$ 0-10 \frac{d i(t)}{d t}-\frac{d^2 i(t)}{d t^2}=0 \\ $$

$$put$$

$$ t=0^{+} \\\ $$

$$ -10 \frac{d i\left(0^{+}\right)}{d t}-\frac{d^2 i\left(0^{+}\right)}{d t^2}=0 \\ $$

$$ -10(100)=\frac{d^2 c\left(0^{+}\right)}{d t^2} \\ $$

$$ \frac{d^2 i\left(0^{+}\right)}{d t^2}=-1000 \mathrm{~A} / \mathrm{s}^2 \\ $$