In a bank, principal increases continuously at the rate of $5\%$ per year. An amount of Rs. 1000 is deposited with this bank, how much will it worth after 10 years $({e^{0.5}} = 1.648)$.

: Let $P$ be the principal at any time $t$. According to the given problem,

$\cfrac{{dP}}{{dt}} = \left( {\cfrac{5}{{100}}} \right) \times P \Rightarrow \cfrac{{dP}}{{dt}} = \cfrac{P}{{20}}$

… (1)
Separating the variables in equation (1), we get
$\cfrac{{dP}}{P} = \cfrac{{dt}}{{20}}$

… (2)
Integrating both sides of equation (2),

we get
$\log P = \cfrac{t}{{20}} + {C_1} \Rightarrow P = {e^{\cfrac{r}{{20}}}}{e^{{C_1}}}$

$\Rightarrow P = C{e^{r/20}}$

(where ${e^{{c_1}}} = C$)

… (3)
Now, $P = 1000$, when $t = 0$

Substituting the values of $P$ and $t$ in (3), we get $C = 1000$.

Therefore, equation (3), gives $P = 1000{e^{r/20}}$

Let ${P_1}$ be the principal and $t = 10$ years.

Then,${P_1} = 1000{e^{1/2}} = 1000 \times 1.648 = {\rm{Rs}}{\rm{. }}1648$