Help, Understanding the Wronskian and Fundamental Solutions

[danerape]

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

 

[mighty2000]

Hello Dane,

Thank you for your questions and for sharing your understanding of the Wronskian. The Wronskian is an important concept in the study of linear differential equations and it is understandable that it can be confusing at times. Let me try to address your questions and provide some clarification.

Firstly, your understanding of the Wronskian is correct. 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. And 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. This is because the Wronskian is a determinant and a non-zero determinant implies that the solutions are linearly independent, which is necessary for a unique solution.

Regarding your question about Abels Theorem, let me provide some context first. Abels Theorem is a result in the theory of differential equations that relates the Wronskian of two solutions to a linear homogenous ode to the solution of the ode itself. It states that the Wronskian of two solutions, y1 and y2, can be expressed as W(y1,y2)(t) = c*e^(-int(p(t))), where c is a constant and p(t) is a function associated with the ode. This constant c is often referred to as the "Abel's constant" and it is not necessarily equal to e^c as you mentioned in your post.

The value of c can be determined by using an initial condition. In other words, if we are given an initial condition of the form y(t0) = y0, where t0 is a specific point in the interval and y0 is a constant, then we can use this condition to solve for c. This is why some websites write Abels theorem as W(y1,y2)(t) = W(y1,y2)(t0)*e^(-int(p(t0))). The Wronskian evaluated at t0, denoted as W(y1,y2)(t0), is equal to the Wronskian evaluated at any other point in the interval, and this is why it is used in the formula for c.

To answer your question about the Wronskian being the same over the entire interval guaranteed by the existence and uniqueness theorem, the answer is yes. This is because