solve the differential equation. assume a, b and k are non zero constants.

dR/dt = kR

i don't understand how to do these problems.

so far i have

dR/R = kdt

ln R = kt + C

am i on the right track? what do i do next?

To solve the given differential equation, which is in the form of a first-order linear homogeneous differential equation, you can use separation of variables method.

Here's how you do it step by step:

Step 1: Separate the variables
Write the differential equation in the form:
dR/R = k*dt

Step 2: Integrate both sides
Integrate both sides of the equation with respect to their respective variables (R on the left side and t on the right side):
∫ dR/R = ∫ k*dt

Integrating the left side:
ln|R| = kt + C1, where C1 is the constant of integration.

Integrating the right side:
∫ k*dt = kt + C2, where C2 is another constant of integration.

Therefore, the equation becomes:
ln|R| = kt + C2

Step 3: Combine the constants
Since C1 and C2 are constants, we can combine them into one constant, C:
ln|R| = kt + C

Step 4: Solve for R
To solve for R, we can exponentiate both sides of the equation to remove the natural logarithm:
e^(ln|R|) = e^(kt + C)

e^(ln|R|) simplifies to R, and e^(kt+C) can be rewritten using the properties of exponents as e^C * e^(kt):
R = e^C * e^kt

The constant e^C can be denoted as another constant, A, so the equation becomes:
R = Ae^kt, where A = e^C

And there you have the solution to the differential equation: R = Ae^kt, with A as a constant.

Note: The constant k represents the rate at which the quantity R is changing over time. It is often referred to as the growth or decay constant, depending on the context of the problem. The constant A determines the initial value of R when t = 0.