Exam 1: Introduction to Differential Equations

The population of a town increases at a rate proportional to its population. Its initial population is 1000. The correct initial value problem for the population, $P ( t )$ , as a function of time, t, is

The values of c for which $y = c$ is a constant solution of $y ^ { \prime } = y ^ { 2 } + 5 y - 6$ are

The differential equation $y ^ { \prime } + 3 y = \sin x$ is

The solution of the initial value problem $y ^ { \prime } = 2 y + x , y ( 1 ) = 1 / 4$ is $y = - x / 2 - 1 / 4 + c e ^ { 2 x }$ , where $c =$

A large mixing tank initially contains 100 gallons of water in which 30 pounds of salt have been dissolved. Another brine solution is pumped into the tank at the rate of 4 gallons per minute, and the resulting mixture is pumped out at the same rate. The concentration of the incoming brine solution is 2 pounds of salt per gallon. If $A ( t )$ represents the amount of salt in the tank at time t, the correct differential equation for A is

The values of m for which $y = e ^ { m x }$ is a solution of $y ^ { \prime \prime } - 6 y ^ { \prime } - 7 y = 0$ are

The values of m for which $y = x ^ { m }$ is a solution of $x ^ { 2 } y ^ { \prime \prime } - 7 x y ^ { \prime } + 12 y = 0$ are

In the previous problem, over a long period of time, the total amount of salt in the tank will approach

The solution of the initial value problem $y ^ { \prime } = 2 y + x , y ( - 1 ) = 1 / 2$ is $y = - x / 2 - 1 / 4 + c e ^ { 2 x }$ , where $c =$

In the falling body problem, the units of acceleration might be

The values of m for which $y = e ^ { m x }$ is a solution of $y ^ { \prime \prime } - 4 y ^ { \prime } - 5 y = 0$ are

The values of c for which $y = c$ is a constant solution of $y ^ { \prime } = y ^ { 2 } + 3 y - 4$ are

In the previous problem, after a long period of time, the temperature of the coffee approaches

The solution of the initial value problem $y ^ { \prime } = 5 y , y ( 1 ) = 3$ is $y = c e ^ { 5 x }$ , where $c =$

The differential equation $y ^ { \prime \prime \prime } + 2 y ^ { \prime \prime } + 3 x y ^ { \prime } - 4 e ^ { x } y = \sin x$ is

The differential equation $y ^ { \prime \prime } + 2 y ^ { \prime } + 3 y = \sin y$ is

The initial value problem $y ^ { \prime } = \sqrt { y ^ { 2 } - 9 } , y \left( x _ { 0 } \right) = y _ { 0 }$ has a unique solution guaranteed by Theorem 1.1 if

The differential equation $y ^ { \prime \prime \prime } + 2 y ^ { \prime \prime } + 3 x y ^ { \prime } - 4 e ^ { x } y = \sin x$ is

The differential equation $y ^ { \prime \prime } + 2 y y ^ { \prime } + 3 y = 0$ is

The values of m for which $y = e ^ { m x }$ is a solution of $y ^ { \prime \prime } - 5 y ^ { \prime } + 6 y = 0$ are