Math 212                               Problem Set #1                                    C.S.Davis

1. Consider the differential equation (DE) below.

            xy'' - 2y' = 0

      a. Determine whether this function, y = cx3, is a solution to the DE for all positive real numbers x and all real arbitrary constants c.

      b. Is y = 5x3 a general solution, particular solution, singular solution or none of the above? Explain.

      c. Is y = 0 a general solution, particular solution, singular solution or none of the above. Explain.

 

2. Solve this first order linear initial value problem (IVP).

            xy' - 2y = 12x3e4x ,    y(1) = 2

 

3. Give the general solution to this DE.

            (y2 sin2(x) + sin2(x)) dx - 8 y sec x dy = 0

 

4. Give the general solution to this DE.

            (4x3e2y - 1/x) dx + (2x4e2y + cos y) dy = 0

 

5. Give the general solution to this DE.

            (6 x sin y + 5 y3) dx + (x2cos y + 3 x y2) dy = 0

 

6. Apply the existence and uniqueness theorem for first order initial value problems to determine whether the IVP below has a unique solution.

            y' = sin x (y - 2)2/3,   y(2) = 5

      a. Make the appropriate test on f(x,y).

      b. Give the dimensions of a rectangle for which the theorem guarantees the existence of a unique solution.

      c. Draw a conclusion.

 

7. Tell whether this DE is homogeneous, and if so, of what degree. If it is homogeneous, find the general solution.

            (x2 - xy + y2) dx - xy dy = 0

 

8. Change this DE to the form of a Bernoulli DE and find the general solution.

            (xy + 2y4) dx - 6x2 dy = 0

 

For the next three problems we will solve the same IVP graphically, numerically, and analytically on the interval [0,1].

9.  a. Use the slope field below to solve the linear DE, y' = -2x + y, graphically. Sketch several members of the family of solutions to the DE.

     b. On the same set of axes, carefully draw and identify the graph of the solution to the IVP,

            y' = -2x + y, y(0) = 1. 

     c. Estimate y(1).

                   

10. Use the Euler method for n = 2 iterations (h = 0.5) to solve the IVP,

            y' = -2x + y, y(0) = 1, numerically on [0,1].

11. a. Solve the IVP above analytically. It's a first order linear.

      b. Find y(1) so that you can see how close the graphical and various numerical methods came to the actual value of y(1).

      c. Produce a small table indicating the three methods used and their results. Write a sentence or two declaring the winners and losers, your personal preference and why it is your choice.

12.    For Clairaut's DE,  y = y'x + 1 - (y')3, 
        a. give the one parameter family of solutions, 
        b. give the singular solution in parametric form,
        c. draw on the page a graph of the singular solution and three or more members of the one parameter family of solutions.