An introduction to differential equations

Express the following statements as differential equations. In each case, you will need to introduce notation to describe the important quantities in the statement, so be sure to clearly state what your notation means.

1. The population of a town grows continuously at an annual rate of 1.25%.

2. A radioactive sample loses mass at a rate of 5.6% of its mass every day.

3. You have a bank account that continuous…

Consider the differential equation . Determine, with justification, whether or not each of the following functions is a solution to the differential equation.

1. $v(t) = 1.5t - 0.25t^2$.

2. $v(t) = 3 + 2e^{-0.5t}$.

3. $v(t) = 3$.

4. $v(t) = 3 + Ce^{-0.5t}$ where $C$ is any constant.

In this problem, we test further what it means for a function to be a solution to a given differential equation.

1. Consider the differential equation . Determine whether the following functions are solutions to the given differential equation.

   1. $y(t) = t + 1 + 2e^t$

   2. $y(t) = t + 1$

   3. $y(t) = t + 2$

2. When you weigh bananas in a scale at the grocery store, the height $h$ of the bananas is described by the differential equation where $k$ is the spring constant…