Chapter 4.1, Problem 20P

(a)

To determine

To show: That $W^{\prime }=\left|\begin{array}{ccc}{y}_1& y_2& y_3\\ y_1{}^{\prime }& y_2{}^{\prime }& y_3{}^{\prime }\\ y_1{}^{\prime \text{}\prime \text{}\prime }& y_2{}^{\prime \text{}\prime \text{}\prime }& y_3{}^{\prime \text{}\prime \text{}\prime }\end{array}\right|$ given that, $W=W\left(y_1,y_2,y_3\right)$.

(b)

To determine

To show: That $W^{\prime }=-p_1\left(t\right)W$ by multiplying the first row by $p_3$, second row by $p_2$ and to add these two rows.

(c)

To determine

To show: That $W\left(y_1,y_2,y_3\right)\left(t\right)=c\exp \left(-{\displaystyle \int p_1\left(t\right)}\right)dt$ and it follows that W is either always zero or nowhere zero on I.

(d)

To determine

To generalize: The above given arguments to the nth order equations $y^{\left(n\right)}+p_1\left(t\right)y^{\left(n-1\right)}+\mathrm{...}+p_n\left(t\right)y=0$ with the solutions $y_1,y_2,\dots ,y_n$ and to establish the Abel’s formula that $W\left(y_1,y_2,\mathrm{...},y_n\right)\left(t\right)=c\exp \left(-{\displaystyle \int p_1\left(t\right)}\right)dt$.