A $5 H$ inductor is placed in series with a $10 \Omega$ resistor. An emf of $5 V$ being suddenly applied to the combination. If the rate ( in watt ) at which energy is delivered by the battery at time equal to one time constant  is x and at the same instant the rate at which energy  ( in Watt )is stored in the inductor is y , then 

At any instant $t$, current in L-R circuit is given as

$i=i_0(1-e^{\raisebox{1ex}{$-t$}\!\left/ \!\raisebox{-1ex}{${\zeta }_L$}\right.})$

Here, $i_0=\frac{E}{R}$ and ${\zeta }_L=\frac{L}{R}$

After one time constant ($t={\zeta }_L$) current in the circuit is

$i=\frac{E}{R}\left(1-\frac{1}{e}\right)=\frac{5}{10}\left(1-\frac{1}{e}\right)=0.316 A$

The rate at which the energy is delivered by the battery is

$x=E_i=\left(5\right)(0.316)=1.58 W$ 

The rate at which energy is stored in the inductor is

$y=\frac{d}{dt}\left(\frac{1}{2}L{i}^2\right)=Li\left(\frac{di}{dt}\right)$

Here, $\frac{di}{dt}=\frac{i_0}{{\zeta }_L}{e}^{-1}=\frac{E}{eL}$   (after one time constant)

Substituting the values, we get

$y=\left(L\right)\left(i\right)\left(\frac{E}{eL}\right)=\frac{Ei}{e}$

$=\frac{5\times 0.316}{2.718}=0.582 W$