Recent content by Malmstrom

  1. M

    Graduate Globally defined and unique solutions.

    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...
  2. M

    Graduate Can all subsequences of a function sequence in C([0,1], R) converge uniformly?

    This is not an exercise. This is a question that rose solving an exercise. Can't solve it.
  3. M

    Graduate Let F and y both be continuous for simplicity. Knowing that:[tex]

    Thanks, I was missing something very easy.
  4. M

    Graduate Can all subsequences of a function sequence in C([0,1], R) converge uniformly?

    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.
    • Malmstrom
    • Thread
    • Convergence Uniform Uniform convergence
    • Replies: 2
    • Forum: Calculus
  5. M

    Graduate Let F and y both be continuous for simplicity. Knowing that:[tex]

    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.
  6. M

    Proving Limit of Sequence: $\sqrt[n]{\frac{2^n}{n!}}=0$

    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...
  7. M

    Proving Limit of Sequence: $\sqrt[n]{\frac{2^n}{n!}}=0$

    It's a little simplified, as I get 2 \sqrt[n]{\frac{1}{n!}} .. but what about this term?
  8. M

    Proving Limit of Sequence: $\sqrt[n]{\frac{2^n}{n!}}=0$

    If you mean e^{\frac{1}{n} \log (\frac{2^n}{n!}) } it's not helping me.
  9. M

    Proving Limit of Sequence: $\sqrt[n]{\frac{2^n}{n!}}=0$

    Not sure what you mean with exponential form.
  10. M

    Proving Limit of Sequence: $\sqrt[n]{\frac{2^n}{n!}}=0$

    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 ...
  11. M

    Prove Limit of Integral Estimate: Zero

    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...
  12. M

    Solving the ODE: x^3y'-2y+2x=0

    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.
  13. M

    Graduate Proving Unique Solution of $y''+e^{-x}f(y)=0$ with $y(0)=y'(0)=0$

    Hi Mark. You were very helpful indeed but I can't figure out what makes the whole thing go wrong and forces y to be constant.
  14. M

    Graduate Proving Unique Solution of $y''+e^{-x}f(y)=0$ with $y(0)=y'(0)=0$

    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.
  15. M

    Graduate Is the Solution to y'=\sqrt{y^2+x^2+1} Defined for All x and Greater Than sinhx?

    Not really, I just don't know where to start from.