PROBLEM 32E

By completing the following steps, prove that the Wronskian of any two solutions y1, y2 to the equation t0 and t in (a, b) , where the constant C depends on y1 and y2.

(a) Show that the Wronskian W satisfies the equation W’+pW = 0.

(b) Solve the separable equation in part (a).

(c) How does Abel’s formula clarify the fact that the Wronskian is either identically zero or never zero on (a,b)?

Step 1:

In this question, we have to show that the Wronskian of any two solutions y1, y2 to the equation

Step 2:

Suppose that y1 and y2 are the two solutions for on the interval (a,b)

a). Wronskian solution is:

Now take the first derivative of equation

We have to prove that

We know that

Proved that