Please help . confused here.

The rate of change of an investment account earning continuous compound interest is given by

dA/dt=kA

where k is a positive constant. The initial account value was $2500. At the end of the third year, the account value was $4200.

Find the particular solution to the differential equation. You may also use A = Pekt.

To find the particular solution to the differential equation, we can use the formula A = Pekt, where A is the account value at time t, P is the initial account value, k is the constant rate of change, and e is the mathematical constant approximately equal to 2.71828.

Given that the initial account value was $2500, we can substitute P = 2500 into the formula:

A = 2500e^kt

Now we need to find the value of k. We are given that the account value was $4200 at the end of the third year, so we can substitute A = 4200 and t = 3 into the equation:

4200 = 2500e^3k

Now we need to solve this equation for k. Divide both sides of the equation by 2500:

1.68 = e^3k

To get rid of the exponential term, we can take the natural logarithm (ln) of both sides:

ln(1.68) = ln(e^3k)

Using the property of logarithms that ln(e^x) = x, we can simplify further:

ln(1.68) = 3k

Now we can solve for k by dividing both sides by 3:

k = ln(1.68) / 3 ≈ 0.2007

Now that we have found the value of k, we can substitute it back into the formula to find the particular solution:

A = 2500e^(0.2007t)

Therefore, the particular solution to the differential equation is A = 2500e^(0.2007t).