Chapter 4.2, Problem 34E

Wronskian. For any two differentiable functions $y_1$ and $y_2$, the function

$W\left[y_1,y_2\right]\left(t\right)=y_1\left(t\right){y^{\prime }}_2\left(t\right)-{y^{\prime }}_1\left(t\right)y_2\left(t\right)$ … $\left(18\right)$

is called the Wronskian of $y_1$ and $y_2$. This function plays a crucial role in the proof of Theorem 2.

Historical Footnote: The Wronskian was named after the polish mathematician H. Wronski $\left(1778-1863\right)$

a. Show that $W\left[y_1,y_2\right]$ can be conveniently expressed as the $2\times 2$ determinant

$W\left[y_1,y_2\right]\left(t\right)=\left|\begin{array}{cc}{y}_1\left(t\right)& y_2\left(t\right)\\ {y^{\prime }}_1\left(t\right)& {y^{\prime }}_2\left(t\right)\end{array}\right|$

b. Let $y_1\left(t\right)$, $y_2\left(t\right)$ be a pair of solutions to the homogeneous equation $a{y}^{″}+b{y}^{\prime }+cy=0$ (with $a\ne 0$) on an open interval $I$. Prove that $y_1\left(t\right)$ and $y\left(t\right)$ are linearly independent on $I$ if and only if their Wronskian is never zero on $I$. [Hint: This is just a reformulation of Lemma 1.]

c. Show that if $y_1\left(t\right)$ and $y_2\left(t\right)$ are any two differentiable functions that are linearly dependent on $I$, then their Wronskian is identically zero on $I$.