Chapter 6.1, Problem 30E

To determine

(a)

To show:

Differentiation of $W\left(x\right):=W\left[y_1,y_2,y_3\right]\left(x\right)$ is

$W^{\prime }\left(x\right)=\left|\begin{array}{rll}{y^{\prime }}_1& {y^{\prime }}_2& {y^{\prime }}_3\\ {y^{\prime }}_1& {y^{\prime }}_2& {y^{\prime }}_3\\ {y^{″}}_1& {y^{″}}_2& {y^{″}}_3\end{array}\right|+\left|\begin{array}{rll}{y}_1& y_2& y_3\\ {y^{″}}_1& {y^{″}}_2& {y^{″}}_3\\ {y^{″}}_1& {y^{″}}_2& {y^{″}}_3\end{array}\right|+\left|\begin{array}{rll}{y}_1& y_2& y_3\\ {y^{\prime }}_1& {y^{\prime }}_2& {y^{\prime }}_3\\ {y^{‴}}_1& {y^{‴}}_2& {y^{‴}}_3\end{array}\right|$

To determine

(b)

To show:

$W^{\prime }\left(x\right)=\left|\begin{array}{ccc}{y^{\prime }}_1& {y^{\prime }}_2& {y^{\prime }}_3\\ {y^{\prime }}_1& {y^{\prime }}_2& {y^{\prime }}_3\\ {y^{″}}_1& {y^{″}}_2& {y^{″}}_3\end{array}\right|+\left|\begin{array}{ccc}{y}_1& y_2& y_3\\ {y^{″}}_1& {y^{″}}_2& {y^{″}}_3\\ {y^{″}}_1& {y^{″}}_2& {y^{″}}_3\end{array}\right|+\left|\begin{array}{ccc}{y}_1& y_2& y_3\\ {y^{\prime }}_1& {y^{\prime }}_2& {y^{\prime }}_3\\ {y^{‴}}_1& {y^{‴}}_2& {y^{‴}}_3\end{array}\right|$

Can be reduced to $W^{\prime }\left(x\right)=\left|\begin{array}{ccc}{y}_1& y_2& y_3\\ {y^{\prime }}_1& {y^{\prime }}_2& {y^{\prime }}_3\\ {y^{‴}}_1& {y^{‴}}_2& {y^{‴}}_3\end{array}\right|$

To determine

(c)

To Show:

The equation $y_i{}^{(3)}\left(x\right)=-{\displaystyle \sum _{k=1}^3{p}_k\left(x\right)y_i{}^{\left(3-k\right)}}\left(x\right)$ for $i=1,2,3$.

To determine

(d)

To show:

$W^{\prime }\left(x\right)=-p_1\left(x\right)W\left(x\right)$ with the help of $W^{\prime }\left(x\right)=\left|\begin{array}{ccc}{y}_1& y_2& y_3\\ {y^{\prime }}_1& {y^{\prime }}_2& {y^{\prime }}_3\\ {y^{‴}}_1& {y^{‴}}_2& {y^{‴}}_3\end{array}\right|$ and $\begin{array}{c}{y}_i{}^{(3)}\left(x\right)=-{\displaystyle \sum _{k=1}^3{p}_k\left(x\right)y_i{}^{\left(3-k\right)}\left(x\right)}\\ \left(i=1,2,3\right)\end{array}$

To determine

(e)

To deduce:

Abel’s identity by solving the first order differential equation.