# Feedback regarding a solution based on a Wronskian

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.
• Thanks again! Then my next question is: how does one check that? It seems nearly impossible to solve this equation, and Wolfram Alpha gives a solution that is so complicated that it does not seem that that is what they intended. I can not obtain any other information about the Wronskian unless I manage to compute $y_1$ and $y_2$, right? Sorry, it's my second week studying Differential Equations, so it is all brand new to me... Jul 23 '19 at 20:01