Take a Cauchy problem like:
y'=(y^2-1)(y^2+x^2)
y(0)=y_0
Show that the problem has a unique maximal solution.
Show that if |y_0| < 1 the solution is globally defined on R whereas if y_0 > 1 it is not.
I'm having trouble with this type of questions: how does one prove global...
Let y_n be a sequence of functions in \mathcal{C}([0,1], \mathbb{R})
Suppose that every subsequence of y_n has a subsequence that converges uniformly. Prove that they all converge to the same limit.
Let F and y both be continuous for simplicity. Knowing that:
\int_0^x F'(t)y^2(t) dt = F(x) \quad \forall x \geq 0
can you say that the function y is bounded? Why? I know that \int_0^x F'(t) dt = F(x) but I can't find a suitable inequality to prove rigorously that y is bounded.
It hopefully goes to zero but this is not a proof. It is not clear at all because for instance
\lim_{n \rightarrow \infty} \sqrt[n]{\frac{1}{n}} = 1
so...
Homework Statement
Prove that
\lim_{n \rightarrow \infty} \sqrt[n]{\frac{2^n}{n!}}=0
Homework Equations
The Attempt at a Solution
Seems really tricky ...
Homework Statement
I have to prove that the solution of an ODE can be continued to a function \in \mathcal{C}^1(\mathbb{R}) . The solution is:
e^{-\frac{1}{x^2}} \int_{x_0}^x -\frac{2e^{\frac{1}{t^2}}}{t^2} dt
It is clear that this function is not defined in x=0 . Its limit for x \rightarrow...
Homework Statement
Solve the following ODE:
x^3y'-2y+2x=0
Homework Equations
The solution should be a function continuous in R \ {0}.
The Attempt at a Solution
Pretty helpless about this one.
This is homework of a past course I did not attend, so I don't *have to* do this stuff. I'm doing it 'cause I'm attending some different stuff about ODEs and lack some of the prerequisites. Anyway I have absolutely no problem in posting any future question in the homework section if you tell me to.