Chapter 6.1, Problem 34E

To determine

To prove:

$\left|\begin{array}{cccc}{f}_1\left(x\right)& f_2\left(x\right)& f_3\left(x\right)& y\\ {f^{\prime }}_1\left(x\right)& {f^{\prime }}_2\left(x\right)& {f^{\prime }}_3\left(x\right)& y^{\prime }\\ {f^{″}}_1\left(x\right)& {f^{″}}_2\left(x\right)& {f^{″}}_3\left(x\right)& y^{″}\\ {f^{‴}}_1\left(x\right)& {f^{‴}}_2\left(x\right)& {f^{‴}}_3\left(x\right)& y^{‴}\end{array}\right|=0$

is a third-order differential equation for which $\left\{f_1,f_2,f_3\right\}$ is a fundamental solution set

and find coefficient of $y^{‴}$