Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

Let $ \phi : \mathbb R \to \mathbb R $ a continuous function. Prove that the initial value problem

$$ y' =y + \phi (x) e^{ -y^{2}} ; \quad y(0)=1$$

has a unique solution on the real line.

(ii) If $ y$ is the unique solution prove that

$$ | y(x) -e^{x} | \leq ( e^x -1) \max_{ 0 \leq t \leq x} | \phi (t) | \quad \forall x>0 .$$


I have prove that there is a unique solution but I can't prove the inequality in (ii).

Any help?

Thanks in advance!

share|improve this question
    
If this is homework, it deserves a homework tag... –  Fabian Jan 7 '13 at 18:22
    
No it is not a homework. That's why there is no homework tag! –  passenger Jan 7 '13 at 18:25

1 Answer 1

up vote 3 down vote accepted

Hints:

  • Write down a differential equation for $u(x) = y(x) - e^x$.

  • For $0\leq t \leq x$, $$\left|\phi(t) e^{-y^2}\right| \leq \max_{ 0 \leq t \leq x} | \phi(x)|. $$

Spoiler below:

$$u' = u + \phi(x) e^{-y^2}, u(0)=0,$$ You can get the estimates for $u(x)$ from the fact that $u'=u + f(x)$ has the solution $$ u(x) =e^{x} \int_0^x e^{-t} f(t) dt $$ here $f(x) = \phi(x) e^{-y^2}$. We have $$\left|\int_0^x e^{-t} f(t) dt\right| \leq \max_{ 0 \leq t \leq x} | \phi(x)| \int_0^x e^{-t} dt = \max_{ 0 \leq t \leq x} | \phi(x)| (1- e^{-x}).$$ Thus, $$ |u(x)| \leq \max_{ 0 \leq t \leq x} | \phi(x)| (e^{x} -1).$$

share|improve this answer
    
Can you explain how did you find $ u_{ \max} $...? –  passenger Jan 7 '13 at 18:40
    
I hope it is now better understandable? –  Fabian Jan 7 '13 at 18:41
    
Thank you very much for your time! Nice solution! Every step in now clear to me! –  passenger Jan 7 '13 at 18:49
    
Just one small question: Is this a general way to attack problems like this one, where we need to estimate the quantity $ | y(x) - e^x|$ ( where $y$ is a solution to a initial valu problem)...? –  passenger Jan 7 '13 at 18:52
    
Yes. See above :-) (write down an equation/solution for $y-e^x$ and estimate) –  Fabian Jan 7 '13 at 18:57

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.