Differential 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?

Re: Difference equation - struggling to solve the equation below

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.

Re: Difference equation - struggling to solve the equation below

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.....

Re: Difference equation - struggling to solve the equation below

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