Let P be a polynomial of degree $n,\,\,\, n\ge 2.$

Question

I am stuck with the following problem:

Let P be a polynomial of degree $n,\,\,\, n\ge 2.$ Then I have to prove that the initial value problem $u'(t)=P(u(t)),\,\,u(0)=1$ has always

1. a unique solution in any interval containing $0$.

2. no solution in any interval containing $0$ for some P.

I have to check which of the aforementioned statements is correct?

My Attempt: Without loss of generality,I take $P(u)=u^n$. Then $u'(t)=P(u(t)),\,\,u(0)=1$ yields to $u^{1-n}=(1-n)t+1.$ Now,if I take $n=2,\,\,$ then we get $\,\,\frac1u=-t+1.$

Now,I am stuck. Can someone help?

Answer 1

I don't think that "without loss of generality,I take $P(u)=u^n$" is a legal move in this game. Try to reason along the following lines:

1. A polynomial function is differentiable, with continuous derivative.
2. A differentiable function with bounded derivative is Lipschitz.
3. The Picard existence/uniqueness theorem applies when the right-hand side is a Lipschitz function of $u$.