Help, Understanding the Wronskian and Fundamental Solutions!

  • Thread starter danerape
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danerape
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Hello,

I have been losing a great deal of sleep trying to understand the Wronskian. Here is the spill on my understanding:

1.The Wronskian of two solutions to a linear homogenous ode can be evaluated at any point within the interval where a unique solution is guaranteed by the existence and uniqueness theorem.

2.If the Wronskian is non-zero at any point on this interval, it automatically implies it is non-zero at every point on the interval.

3.These two statements above prove c1y1+c2y2 to be the general solution because a non-zero Wronskian implies any IVP can be solved for c1 and c2.

My confusion is specifically with Abels Theorem. When you prove Abels Theorem, the constant you get upon integrating is e^c.

In other words, Abels theorem states W(y1,u2)(t)=ce^(-int(p(t)))

Where c, the constant is equal to e^c.

How can the value of c ever be 0?

I am also confused on how c is determined in an application of the theorem. I have seen some web sites write Abels theorem like this....

W(y1,y2)(t)=W(y1,y2)(t0)e^(-int(p(t0))

What?

Does this mean that c is the Wronskian evaluated at a particular point t0? Is this the point chosen by an initial condition? Does this mean the Wronskian is the same over the entire interval guaranteed by the existence and uniqueness theorem? Does this mean that no matter which point t0 I choose to evaluate the Wronskian at, the Wronskian will be the same?

Please elaborate,

Thanks,

Dane
 

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