Chapter 21, Linear, First-Order Differential Equations Video Solutions, Mathematics for Economics

Problem 1

Solve the following linear, first-order differential equations, and ensure that the initial conditions are satisfied:
(a) $\dot{y}-y=0$ and $y(0)=1$
(b) $\dot{y}+3 y=12$ and $y(0)=10$
(c) $2 \dot{y}+\frac{1}{2} y=12$ and $y(0)=10$
(d) $\dot{y}=5$ and $y(0)=1$
(e) $\dot{y}=6 y-6$ and $y(0)=3$

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Problem 2

Solve the following linear, first-order differential equations, and ensure that the initial conditions are satisfied:
(a) $10 \dot{y}=5 y$ and $y(0)=1$
(b) $4 \dot{y}-4 y=-8$ and $y(0)=10$
(c) $\dot{y}=7$ and $y(0)=2$
(d) $\dot{y}=2 y-1$ and $y(0)=5$
(e) $\dot{y}+2 y=4$ and $y(0)=3$

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02:15

Problem 3

Let $p(t)$ represent the consumer price index. If the rate of inflation of the price index is constant at $5 \%$ (i.e., the growth rate of $p(t)$ is $5 \%$ ), and the price index has a base value of 100 at time $t=0$, solve for the expression showing the price index as a function of time.

Foster Wisusik

Numerade Educator

00:00

Problem 4

If income per capita is growing at a constant rate of $3 \%$, how long will it take for it to double?

Zewei Lei

Lehigh University

03:59

Problem 5

Let $y(t)$ be the reserves of oil left in an oil pool at time $t$. Suppose that extraction reduces reserves at a constant rate equal to $\alpha$. (The rate of decline of reserves is $\alpha$.) If initial reserves at $t=0$ were 500 million barrels, solve for the expression showing reserves as a function of time.

Amit Srivastava

Numerade Educator

03:19

Problem 6

Use the information in exercise 5 and assume that $\alpha=0.1$. Find the time at which $50 \%$ of oil reserves have been used up. Find the time when $95 \%$ of oil reserves are used up.

Darshan Maheshwari

Numerade Educator

01:24

Problem 7

On the floor of the stock exchange, traders meet to buy and sell stock in various companies. Suppose that the change in the quantity sold of a particular stock depends on the gap between the offer price, $p^p$, and the asking price, $p^5$. In particular, assume that $\dot{q}=\alpha\left(p^D-p^5\right)$. The inversedemand function of the buyers is

$$
p^D=a+b q
$$

and the inverse-supply function of the sellers is

$$
p^s=g+h q
$$

If initial price is $p_0$ at $t=0$, find the equilibrium quantity sold in this market and the expression showing quantity sold as a function of time. What conditions on the parameters of the inverse demand and supply curves must hold for the equilibrium to be stable?

Carson Merrill

Numerade Educator