I. Solve the following separable differential equations. Find the general solution if no initial condition is present. These problem are based on Section 2.2

1. y' = y and y(0)=-6

2. y' = y and y(1) = e²

3. y' = cos(2x) + 2 and y(pi) = 8

4. y' = y²e^x and y(0)=2

5. y' = e^x+y

6. y' = x²cos²(y)

7. y' = y² e^x

8. y' = x sin(x) cot(y) and y(pi/4)=6

9. y' = x³y² + xy² + 2y² x³ + x + 2 and y(0) = pi/4

10. y' = xy³sqrt(1+x²) and y(0)=1

II. Describe the set of initial conditions for which each differential equation below is guaranteed to satisfy Theorem 2.4.2. Do not try to solve them.

1. y' = (x+2y)/(2-y)

2. y' = x³/(x+2y)

3. y' = y/(x-3) + x²/(y+4)

4. y' = tan(x)/(y²-1)

5. y' = cot(xy)

6. y' = sqrt(y)/(x+y³)

7. y' = (xy)^2/3 + ln(xy)

III. Draw the direction field by hand for the following differential equations. Use graph paper with grid locations from -5 to 5 for x and y. Do not solve these equations.

1. y' = 2x-1

2. y' = (x/2) + 2

3. y' = y+1

4. y' = 2x+y

5. y' = (1/3)(x²-1)

6. y' = y e^x

7. y' = 3x-y

8. y' = y-x

9. y' = x²-4

10. y' = (3y+1)/x