According to Problem of Section 2.1, the Wronskian W(y1, y2) of two solutions of the secon...

According to Problem of Section 2.1, the Wronskian W(y1, y2) of two solutions of the second-order equation

y″ + P1(x)y′ + P2(x)y = 0

is given by Abel’s’s formula

for some constant K. It can be shown that the Wronskian of n solutions y1, y2,..., y„ of the nth-order equation

satisfies the same identity. Prove this for the case 11 = 3 as follows:

(a) The derivative of a determinant of functions is the sum of the determinants obtained by separately differentiating the rows of the original determinant. Conclude that

(b) Substitute for y1(3), y2(3), and y3(3) from the equation

y(3) + p1y″ + p2y′ + p3y = 0,

and then show that W′ = −p1W. Integration now gives Abel’s formula.

Problem

Prove that an nth-order homogeneous linear differential equation satisfying the hypotheses of Theorem 2 has n linearly independent solutions y1, y2,..., yn. (Suggestion: Let yi be the unique solution such that

y1(i−l)(a) = 1 and y1(k) a = 0 if k = i −1.