Exercises

After completing the tutorial attempt the following exercises.

If you are not sure how to do something, have a look at the “How To” section.

1. Create the following differential equations:

   1. \(\frac{dy}{dx} = \cos(x)\)

   2. \(\frac{dy}{dx} = 1 - y\)

   3. \(\frac{dy}{dx} = \frac{x - 50}{10}\)

   4. \(\frac{dy}{dx} = y ^2 \ln (x)\)

   5. \(\frac{dy}{dx} = (1 + y) ^ 2\)

2. Obtain the general solution for the equations in question 1.

3. Obtain the particular solution for the equations in question 1 with the following particular conditions:

   1. \(y(0) = \pi\)

   2. \(y(2) = 3\)

   3. \(y(50) = 1\)

   4. \(y(e) = 1\)

   5. \(y(-1) = 3\)

4. The rate of increase of a population (\(p\)) is equal to 1% of the size of the population.

   1. Define the differential equation that models this situation.

   2. Given that \(p(0)=5000\) find the population after 5 time units.

5. The rate of change of the temperature of a hot drink is proportional to the difference between the temperature of the drink (\(T\)) and the room temperature (\(T_R\)).

   1. Define the differential equation that models this situation.

   2. Solve the differentia equation.

   3. Given that \(T(0) = 100\) and the room temperature is $\(T_R=20\)$ obtain the particular solution.

   4. Use the particular solution to identify how on it will take for the drink to be ready for consumption (a temperature of 80) given that after 3 time units \(T(3)=90\).