Difference between revisions of "Aufgaben:Problem 14"

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(Created page with "=Problem 4a= Let \( T > 0,\eta \in C^{1}([0,T]), \phi \in C^{0}([0,T]) \), and assume that: $$ \frac{d}{dt}\eta(t)<=\eta(t)\phi(t), \forall t \in [0,T] $$ Show that: $$\eta(t...")
 
(Problem 4a)
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=Problem 4a=
 
Let \( T > 0,\eta \in C^{1}([0,T]), \phi \in C^{0}([0,T]) \), and assume that:
 
$$ \frac{d}{dt}\eta(t)<=\eta(t)\phi(t), \forall t \in [0,T] $$
 
  
Show that:
 
$$\eta(t)<=\eta(0)e^{\int_{0}^{t} \phi(s) ds}, \forall t \in [0,T] $$
 
 
Hint: consider $$ \chi(t):= \eta(t) e^{-\int_{0}^{t} \phi(s) ds $$
 

Revision as of 14:39, 26 December 2014