Exercices equations differentielles

[Anonyme]

Bon j'ai 2 exercices sur les equations différentielles ...mais je suis
completement bloqué...
Merci à ceux qui pourrait me donner les réponses pour que j'arrive
enfin à comprendre !

Exercice 1

Let x be wealth and let u(x) be an individual's utility function,
depending on wealth.
Define the function µ(x) = - u"(x)/u'(x).
Then µ(x) is called the Arrow-Pratt (AP) measure of absolute risk
aversion.
Consider a function that has constant AP absolute risk aversion given
by K

a)Obtain a second order differential equation for utility, u(x).
b) Solve the equation to obtain a functionnal form for utility
functions that have constant AP absolute risk aversion
c) Show that this soltion satisfies the second order differential
equation
d)Sketch the shape of your solution where k >0 and u'(x) >0



Exercice 2

The market for a product is given by the following demand and supply
equations.
qd(t) = 500- p(t) -p'(t)
qs(t)= 3p(t) + 1/2 p'(t) -16

Assume that the market always clears.

a)Obtain a second order differential equation for prices
b)Solve the differential equation assuming initial values p(0)=20,
p'(0)=10
c)Determine whether or not process converges to equilrium
d) What is the frequency of oscillations about equilibrium?


Merci beaucoup ...c'est très important !

[Anonyme]

"Yo666" a écrit dans le message de news:
49ee0bb3.0502110941.6060bf8b@posting.google.com...
> Bon j'ai 2 exercices sur les equations différentielles ...mais je suis
> completement bloqué...
> Merci à ceux qui pourrait me donner les réponses pour que j'arrive
> enfin à comprendre !
>
> Exercice 1
>
> Let x be wealth and let u(x) be an individual's utility function,
> depending on wealth.
> Define the function µ(x) = - u"(x)/u'(x).
> Then µ(x) is called the Arrow-Pratt (AP) measure of absolute risk
> aversion.
> Consider a function that has constant AP absolute risk aversion given
> by K

Je crois comprendre que :
K = - u"(x)/u'(x)
soit :
d(u'(x))/u'(x) = -K.dx
d'où en intégrant :
u'(x) = C.exp(- K.x)
u(x) = - C.exp(- K.x)/K + C'
C et C' étant des constantes

>
> a)Obtain a second order differential equation for utility, u(x).
> b) Solve the equation to obtain a functionnal form for utility
> functions that have constant AP absolute risk aversion
> c) Show that this soltion satisfies the second order differential
> equation
> d)Sketch the shape of your solution where k >0 and u'(x) >0
>
>
>
> Exercice 2
>
> The market for a product is given by the following demand and supply
> equations.
> qd(t) = 500- p(t) -p'(t)
> qs(t)= 3p(t) + 1/2 p'(t) -16
>
> Assume that the market always clears.

là, il faudrait savoir ce que l'on veut dire ?
Probablement quelquechose qui relie qd(t) et qs(t) mais quoi ?
ce n'est plus des maths ; il faudrait d'abord poser le problème
clairement....

>
> a)Obtain a second order differential equation for prices
> b)Solve the differential equation assuming initial values p(0)=20,
> p'(0)=10
> c)Determine whether or not process converges to equilrium
> d) What is the frequency of oscillations about equilibrium?
>
>
> Merci beaucoup ...c'est très important !

[Anonyme]

"A.J." a écrit dans le message de news:
420e1542$0$802$8fcfb975@news.wanadoo.fr...
>
> "Yo666" a écrit dans le message de news:
> 49ee0bb3.0502110941.6060bf8b@posting.google.com...[color=green]
>> Bon j'ai 2 exercices sur les equations différentielles ...mais je suis
>> completement bloqué...
>> Merci à ceux qui pourrait me donner les réponses pour que j'arrive
>> enfin à comprendre !
>>
>> Exercice 1
>>
>> Let x be wealth and let u(x) be an individual's utility function,
>> depending on wealth.
>> Define the function µ(x) = - u"(x)/u'(x).
>> Then µ(x) is called the Arrow-Pratt (AP) measure of absolute risk
>> aversion.
>> Consider a function that has constant AP absolute risk aversion given
>> by K
>
> Je crois comprendre que :
> K = - u"(x)/u'(x)
> soit :
> d(u'(x))/u'(x) = -K.dx
> d'où en intégrant :
> u'(x) = C.exp(- K.x)
> u(x) = - C.exp(- K.x)/K + C'
> C et C' étant des constantes
>
>>
>> a)Obtain a second order differential equation for utility, u(x).
>> b) Solve the equation to obtain a functionnal form for utility
>> functions that have constant AP absolute risk aversion
>> c) Show that this soltion satisfies the second order differential
>> equation
>> d)Sketch the shape of your solution where k >0 and u'(x) >0
>>
>>
>>
>> Exercice 2
>>
>> The market for a product is given by the following demand and supply
>> equations.
>> qd(t) = 500- p(t) -p'(t)
>> qs(t)= 3p(t) + 1/2 p'(t) -16
>>
>> Assume that the market always clears.
>
> là, il faudrait savoir ce que l'on veut dire ?
> Probablement quelquechose qui relie qd(t) et qs(t) mais quoi ?
> ce n'est plus des maths ; il faudrait d'abord poser le problème
> clairement....[/color]

Si cela voulait dire que les courbes qd(t) et qs(t) ont la même pente, c'est
à dire :
(qd(t))' = (qs(t))'
on dériverait et égalerait les 2 seconds membres, soit :
3.p" + 8.p' = 0
et on retombe sur une solution analogue à celle de l'exercice 1.
Mais cette solution n'est pas périodique....
[color=green]
>>
>> a)Obtain a second order differential equation for prices
>> b)Solve the differential equation assuming initial values p(0)=20,
>> p'(0)=10
>> c)Determine whether or not process converges to equilrium
>> d) What is the frequency of oscillations about equilibrium?
>>
>>
>> Merci beaucoup ...c'est très important !
>
>[/color]