Question

Consider the pair of functions $y_1 = t$ and $y_2 = 2t^2$. Which of these statements are true? Select all that apply. $W[y_1, y_2](t) = 2t^2$ $W[y_1, y_2](t) > 0$, for all values of t in the interval $(-3, 3)$. Abel's theorem implies that $y_1$ and $y_2$ cannot both be solutions of any differential equation of the form $y'' + p(t)y' + q(t)y = 0$ on the interval $(-3, 3)$. The pair $y_1$ and $y_2$ constitutes a fundamental set of solutions to some second- order differential equation of the form $y'' + p(t)y' + q(t)y = 0$ on the interval $(-3, 3)$. Since there exists a value of $t_0$ in the interval $(-3, 3)$ for which $W[y_1, y_2](t_0) = 0$, there must exist a differential equation of the form y'' + p(t)y' + q(t)y = 0 for which the pair $y_1$ and $y_2$ constitutes a fundamental set of solutions on the interval $(-3, 3)$.

          Consider the pair of functions $y_1 = t$ and $y_2 = 2t^2$.
Which of these statements are true? Select all that apply.
$W[y_1, y_2](t) = 2t^2$
$W[y_1, y_2](t) > 0$, for all values of t in the interval $(-3, 3)$.
Abel's theorem implies that $y_1$ and $y_2$ cannot both be solutions of any differential
equation of the form $y'' + p(t)y' + q(t)y = 0$ on the interval $(-3, 3)$.
The pair $y_1$ and $y_2$ constitutes a fundamental set of solutions to some second-
order differential equation of the form $y'' + p(t)y' + q(t)y = 0$ on the interval
$(-3, 3)$.
Since there exists a value of $t_0$ in the interval $(-3, 3)$ for which
$W[y_1, y_2](t_0) = 0$, there must exist a differential equation of the form
y'' + p(t)y' + q(t)y = 0 for which the pair $y_1$ and $y_2$ constitutes a fundamental
set of solutions on the interval $(-3, 3)$.

Consider the pair of functions y1 = t and y2 = 2t^2.
Which of these statements are true? Select all that apply.
W[y1, y2](t) = 2t^2
W[y1, y2](t) > 0, for all values of t in the interval (-3, 3).
Abel's theorem implies that y1 and y2 cannot both be solutions of any differential
equation of the form y” + p(t)y' + q(t)y = 0 on the interval (-3, 3).
The pair y1 and y2 constitutes a fundamental set of solutions to some second-
order differential equation of the form y” + p(t)y' + q(t)y = 0 on the interval
(-3, 3).
Since there exists a value of t0 in the interval (-3, 3) for which
W[y1, y2](t0) = 0, there must exist a differential equation of the form
y” + p(t)y' + q(t)y = 0 for which the pair y1 and y2 constitutes a fundamental
set of solutions on the interval (-3, 3).

Added by Jonathan W.

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