hi, how can i start to solve?

If you an indefinite integral, do you have information that lets you determine the constant?
 
this is the whole question. there is no other given, like a number. And our book is fundamental methods of mathematical economics + an introduction to differential equations and their applications
 
s gonzales said:
this is the whole question. there is no other given, like a number. And our book is fundamental methods of mathematical economics + an introduction to differential equations and their applications
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Yes - there is condition given.

Please rewrite - not copy/paste or image - part (a) of your question again!
 
You probably want to know what I understand. but I did not understand what the problem was to tell us. the question is a weird a bit. we know S is decreasing with (t) and question wants from us initial value. sorry i am not clever student.
 
I saw something like this about general solution.
dy/dt + ay =b
y(t) =y(complementary) + y(particular) =Ae^-at + b/a
A =y(0)
 
s gonzales said:
You probably want to know what I understand. but I did not understand what the problem was to tell us. the question is a weird a bit. we know S is decreasing with (t) and question wants from us initial value. sorry i am not clever student.
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No - I do not (yet) want to know what you understand

Please rewrite - not copy/paste or image - part (a) of your question again!

If you do not understand this problem - I do not know why are you taking this class?!!
 
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s gonzales said:
this is the whole question. there is no other given, like a number. And our book is fundamental methods of mathematical economics + an introduction to differential equations and their applications
Click to expand...
To supplement Subhotosh's approach, you need to understand the fundamental fact that, in differential equations, the unknown is a member of a set of one or more functions, not a number. You frequently need additional information to determine which function of the set is the one you need. Only after you know what function you are dealing with can you plug in numbers. It is a multi-step process.

Furthermore, you may not know what the dot notation means. It means

[MATH]\dot S(t) \equiv \dfrac{dS}{dt}.[/MATH]
 
this is why? it is a rule? did you want the question as a text instead of image
a-find for St when sales at time 0.
 
Did you do the integration that SK suggested? If so, what did you get.
 
[MATH]\int \dfrac{ds}{S} = \int -a \ dt \implies\\ ln(S) + c_1 = - at + c_2 \implies ln(S) = - at + c,\\ \text {where } c = c_2 - c_1.\ \checkmark[/MATH]Now you have a set of functions, differing by the value of c. Can you determine c? Well, what else do you know?

[MATH]S(0) = S_0.[/MATH] You are given that in the statement of the problem, correct?

[MATH]\text {Therefore, it must be true that } ln(S_0) = -a(0) + c \implies\\ c= ln(S_0).[/MATH]Is that not obvious?

[MATH]ln(S) = - at + c = -at + ln(S_0) \implies\\ - at = \ln(S) - ln(S_0) = ln \left ( \dfrac{S}{S_0} \right ) \implies\\ \dfrac{S}{S_0} = e^{-at} \implies S = S_0 * e^{-at}.[/MATH]There must be something in your text that talks about separable differential equations. But you just solved one even if you did not understand the text..

Now where is your problem with part b?
 
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s gonzales said:
thank you, everything was very clear. S0 eliminated e^-at=0.5 what will i find
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Yes, in parrt b, you divide by S_0 and are left with

[MATH]e^{-at} = 0.5[/MATH].

Are you asking how to solve that for t?
 
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