IVP and Existence/Uniqueness Problems

[dorado29]

I'm working on a problem for my diffEQ's class. It reads:


For each of the given initial value problems, what is the region where the existence and uniqueness of the solution is guaranteed? Is it all t, or t > -3, or a neighborhood of t = 1?

a) x'' = t x x' + sin(t)
b) x'' = tx + 2x' + sin(t)
c) (t+3)x'' = tx +2x' + sin(t)

The initial values given for a, b and c are all x(1) = 5 and x'(1) = 2



My thoughts/ attempts:

It seems that a, b and c are all linear. In my book (my professor wrote it), it says that

~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
If f(t, x, x') is a continuously differentiable function, then the initial value problem has exactly one solution for t close to t₀. Furthermore, this solution exists for all t, if x'' = f(t, x, x') is a linear function.
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

I don't see any x or x' in a, b, or c that aren't to the first power (I believe that would make them all linear?). The way I'm interpreting the above bit from the book, it would seem that the existence is guaranteed for all t values. I don't think this is right because when equation c is reordered to get x'' on the left and everything else to the right, there is a (t + 3) term in the denominator. Would this mean it's guaranteed for any value of t other than -3 to avoid a 0 denominator value?

I'm unsure if multiplying x and x' together like in equation a would make it nonlinear..

Also, what do you guys think "or a neighborhood of t = 1" means? The word neighborhood throws me off..

 

[HallsofIvy]

First, I am moving this to the "homework" section.

Second, (a) is non-linear because of the "txx'" term- the "x times x'" is considered a non-linear function of x. The others are linear.

A "neighborhood" of a point is any open set that contains the point. If you are not familiar with the term "open set", you can instead think "open interval"- an interval that does not contain its endpoints.

 

[dorado29]

when doing these, am I just concerned with x and x' at the specified points? Would I just plug 5 in where there's an x and 2 when there's an x'? I have no clue how to solve these.. There's no answer key and this is one of the few topics he didn't include an example problem in the book for. Am I trying to find a value of x''?