## anonymous 5 years ago suppose that customer demand depends upon the price trend according to the formula q=60-20p-7p'(t)+p"(t). if the supply function is qs (p)=-12+10p, write down the condition for equilibrium and determine equilibrium price p(t) when p(0)=5 and p'(0)=17

1. anonymous

I'm not trained in economics, just maths/physics. I assume the condition for equilibrium is supply = demand, so$-12+10p=60-20p-7p'+p''$is what you have to solve. Clean this thing up to get,$p''-7p'-30p+72=0$To make this homogeneous, set$p=-\frac{v-72}{30}\rightarrow p'=-\frac{v'}{30} \rightarrow p''=-\frac{v''}{30}$and sub into the differential equation to eliminate p. After doing this, mutiply through by -30 to give,$v''-7v'-30v=0$This is second order, ordinary differential, which can be solved assuming solutions for the form$v=e^{\lambda t}$Substituting this assumed solution into the d.e. yields the characteristic function,$\lambda^2-7\lambda-30\lambda=0\rightarrow \lambda =10, -3$

2. anonymous

So your solution for v is$v=c_1e^{10t}+c_2e^{-3t}$But from our transformation, $v=72p-30$so$72-30p=c_1e^{10t}+c_2e^{-3t} \rightarrow p=c_1e^{10t}+c_2e^{-3t}+\frac{72}{30}$

3. anonymous

(note the constants have absorbed any other constants we've come across in solving for p). You should be able to do the rest. If not, let me know.

4. anonymous

To find the constants,$p(0)=5=c_1+c_2+\frac{72}{30}\rightarrow c_1+c_2=-\frac{13}{5}$

5. anonymous

$p'(0)=17=10c_1-3c_2$From the first condition,$c_2=-\frac{13}{5}-c_1$and subbing into the second condition,$17=10c_1-3(-\frac{13}{5}-c_1)\rightarrow c_1=\frac{46}{65}$and so$c_2=-\frac{13}{5}-\frac{46}{65}=-\frac{43}{13}$

6. anonymous

$p=\frac{46}{65}e^{10t}-\frac{43}{13}e^{-3t}+\frac{72}{30}$

7. anonymous

Hi I tried working it out last night my answer was similar to yours just a few differences in the method of working it out but thank you so much for your insight. I am now starting differential equations and i'm having a hard time catching on...:(

8. anonymous

No probs. It's just experience in recognition with these things (and some insight). If you could 'fan' me as thanks, I'd appreciate it - I need points :)