Difference equation - struggling to solve the equation below

Hi,

I can't quite understand this differential equation question..... any help would be much appreciated:


Imagine a country that is adopting foreign technology T as shown in the following equation where A denotes the level of domestic technology and the dot over A denotes its derivative with respect to time:

(A_t ) ̇/A_t = ϕ(E) [(T_t - A_t)/A_t ] ϕ(0)=0 ϕ'(E)>0



a) Assume that the level of foreign technology is constant and solve the differential equation above


b) What is the effect of an increase in the level of education E?


c) What is the effect of an increase in the rate of growth of foreign technology λ?

Your way of writing this is awkward but I think you mean
$\displaystyle \frac{A'}{A}= \phi(E)\frac{T(t)- A(t)}{A(t)}$.

However, since the ' indicates the derivative and there is no "t+1" or other time change, this is NOT a "difference equation" so perhaps that is not what you mean. Assuming $\displaystyle \phi(E)$ is a given function, we can rewrite this as $\displaystyle \frac{dA}{T- A}= \phi(E)dt$.

Without knowing both $\displaystyle \phi(E)$ and $\displaystyle E(t)$ the best we can do is write the solution to that equation as $\displaystyle T- A= Ce^{-\int \phi(E(t))dt}$ so that $\displaystyle A= T- Ce^{-\int\phi(E(t))dt}$.

You should be able to answer your questions from that.

Hi - sorry - I attempted to write it as a proper equation but it didn't paste properly.....

What I was trying to write was exactly as you have written it EXCEPT with a . over the A in the numerator on the left hand side, not a '.

I have however realised it is a DIFFERENTIAL equation rather than a DIFFERENCE equation..... sorry.

Any further help would be great.....

Whether you used . or ' is not really relevant- it means the derivative, doesn't it?