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.