Assignment For Section 1.4 & 1.5

Readings Questions:

You should read the section before attempting any reading questions or problems. Remember that you are responsible for all definitions and examples. Also remember to keep the answers to the reading questions on a separate piece of paper; these will be randomly collected from time to time.

1)   What is a separable differential equation? Go back through the examples in section 1.1 and determine which of these are separable. Solve each separable example using the method described in section 1.4.

2)   The middle of page 32 says that the graph of a differentiable solution cannot have a vertical tangent line.
        a) Why not?
        b) Why does this force the solution of the initial value problem (given below) to lie on the lower "half" of the oval contour in Fig. 1.4.2 (see Remark 2 on page 32).

3)    a) What is an implicit solution? Give an example.

        b) You saw the word implicit back in Calc 1; what is implicit differentiation, and how does this use of implicit relate to implicit solutions?

4)   What is a singular solution? Give two examples.

5)   Verify by substitution that the function y(x) given in Eq (6) on page 45 satisfies Eq (3) on page 44.

6)   What is an integrating factor?

7)   What is an equilibrium solution?

8)   Why does 1.5 Theorem 1 only guarantee that the solution of example 3 (pg 49) is defined on the positive x-axis? (Why not the whole x-axis?)

9) Solve example 3 with MAPLE:

                            > restart:
                            > with (DEtools):
                               de1:=x^2*diff(y(x),x)+x*y(x)=sin(x);
                            > dsolve({de1,y(1)=y[0]},y(x));

    Does MAPLE contain the Si(x) function?

Problems:

page 40     1, 4, 7, 12, 17, 22, 27, 31, 34, 35, 37, 44, 48

page 52     1, 5, 8, 12, 17, 22, 29, 32, 33, 36, 39