Question

If P dollars are left in the bank at interest I percent per year compounded continuously, find the amount A at time t. Hint: Find dA, the interest on A dollars for time dt.

Short answer

Answer

The amount of money in the bank at any time t is $P=e^{It/100}$if the interest is $I/100$per year.

Step-by-step solution

Given information

The annual interest is $I/100$ if P dollars are invested in the bank.

Definition of ordinary differential equation

An ordinary differential equation (ODE) is a differential equation that contains one or even more functions of one independent variable and derivatives.

Find the amount A at a time t. 

The rate at which the amount of money increasing is proportional to the amount at any time.

$\frac{dA}{dt}=\frac{I}{100}A$

Solve the above differential equation using the variable separable method.

$$\begin{gathered} \int \frac{dA}{A}=\int \frac{I}{100}dt \\ In\left(A\right)=\frac{I}{100}t+C \end{gathered}$$

The general solution for the above differential condition is $A=C_1{e}^{It/100}$, where $C_1$is the integration constant and the initial condition is at the beginning $\left(t=0\right)$the amount of money left in the bank is P.

Use the initial condition, such as at a time $t=0$to find the value of the constant of integration.

Thus, $C_1=C$

Hence, the amount of money at any time is $A=P{e}^{It/100}$.

Therefore, the amount of money in the bank at any time t is $P=e^{It/100}$if the interest is $I/100$per year.