First Order Differential Equation and The Equilibrium Solution Questions

I don’t know how to handle this Calculus question and need guidance.

1)  Solve the initial value problem.

2)  Given

a)  Solve the first order differential equation.

b)  Find the equilibrium solution.

c)  Make a sketch of the direction field, for ,  and determine whether the equilibrium solution is stable.

3)  Consider the following pairs of differential equations that model a predator prey system with populations x and y.  

a)  Find the lines along which .Find the lines along which

b)  Find the equilibrium points for the system.  

c)  Identify the four regions in the first quadrant of the xy-plane in which are positive or negative.

d)  Sketch a representative solution curve in the xy-plane and indicate the direction in which the solution evolves.

4)  Determine the radius and interval of convergence of the following power series.  

5)  Given

a)  Use the definition of a Taylor/Maclaurin series to find the first four nonzero terms of the Taylor series for the given function centered at a.

b)  Write the power series using summation notation.

6)  Evaluate the following limit using Taylor series.  

7)  Write the first three terms of the Maclaurin series for the function