(a) We solve it using the formula:

RL7

We have:

MATH

5_RLex2__3.png

(b) At $t=0.5,$ MATH A

(c) MATH

MATH

(d)

MATH

MATH V

The graph of $V_{R}$ and $V_{L}$:

5_RLex2__13.png

TC for this example is: MATH

NOTE (just for interest): If we could not use the formula in (a), and we did not use separation of variables, we could recognise that the DE is 1st order linear and so we could solve it using an integrating factor.

We use the formula: MATH

The required DE is: MATH

MATH

I.F. = MATH

MATH

Since $i(0)=0$, we have $K=-2$.

So $i=2(1-e^{-5t})$

It works :-)