Draw a slope field for each of the following differential equations.

1. dy / dx = x + 3 2. dy / dx = 4y

3. dy / dx = 2x + 3y 4. dy / dx = 5x

5. dy / dx = y – 4 6. dy / dx = -y / x

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19. The slope field for the differential equation dy / dx = x is shown in the figure below.

a) Find a particular solution y = f(x) to the given differential equation with initial condition f(0) = 3. Graph your solution.

b) Use your solution to find f(0.6).

c) Find the domain of the solution you found in part (a).

d) Find the interval where your solution in part (a) is increasing. Justify your answer.

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20. The slope field for the differential equation dy / dx = 4 – y is shown in the figure below.

a) Find a particular solution y = f(x) to the given differential equation with initial condition f(0) = 2. Graph your solution.

b) Use your solution to find f(1.5).

c) Describe what is happening to the graph at the line y = 4. Explain.

d) Find the interval where your solution in part (a) is concave up / down. Justify your answer.

21. The slope field for the differential equation dy / dx = -x / y is shown in the figure below.

a) Find a particular solution y = f(x) to the given differential equation with initial condition f(0) = -4. Graph your solution.

b) Use your solution to find f(2.1).

c) Where are the foci of your solution found in part (a).

d) Find the range of your solution in part (a).

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