Function $y_1(x)$ and $y_2(x)$ are solutions to the equation $y''+\frac{1}{x}y'+(1-\frac{1}x^2)y=0$ in the open interval $(0,\infty$). It is given that the following conditions are fulfilled:
$y_1(1)=5.5$
$y'_1(1)=3.3$
$y_2(1)=3.5$
$y'_2(1)=2.1$
Please conclude, if possible, whether the functions $y_1(x)$ and $y_2(x)$ are linearly dependent or independent.
I simply looked at the value of the Wronskian, which is positive: because it refers to two functions that are the two solutions to a second order linear homogenous differential equation, I took that to mean that the Wronskian does not equal zero anywhere in the given interval. And therefore the functions are linearly independent.
I am not sure whether this interpretation is correct, as the textbook definitions are not 100% clear to me.
Thank you!
Hint: check whether the Wronskian is zero or non-zero anywhere in the given interval. If it is zero at any point in the interval, then $y_1$ and $y_2$ are linearly dependent, otherwise they are independent.