Differential Equations Answers

1. A large tank is filled to capacity with 1000 L of pure water. Brine containing 0.25 kg of salt per liter is pumped into the tank at a rate of 10 L/min. The well-mixed solution is pumped out at the same rate. Find the number A(t) of kilograms of salt in the tank at time t.

A(t) = ___ kg

2. A tank contains 250 liters of fluid in which 10 grams of salt is dissolved. Brine containing 1 gram of salt per liter is then pumped into the tank at a rate of 5 L/min; the well-mixed solution is pumped out at the same rate. Find the number A(t) of grams of salt in the tank at time t.

A(t) = ___

1. A tank contains 200 liters of fluid in which 20 grams of salt is dissolved. Brine containing 1 gram of salt per liter is then pumped into the tank at a rate of 5 L/min; the well-mixed solution is pumped out at the same rate. Find the number A(t) of grams of salt in the tank at time t.

A(t) = ____ g

2. A tank contains 420 liters of fluid in which 30 grams of salt is dissolved. Pure water is then pumped into the tank at a rate of 6 L/min; the well-mixed solution is pumped out at the same rate. Find the number A(t) of grams of salt in the tank at time t.

A(t) = ___ g

The population of a town grows at a rate proportional to the population present at time t. The initial population of 500 increases by 15% in 10 years. What will be the population in 50 years? (Round your answer to the nearest person.)

____ persons

How fast is the population growing at t = 50? (Round your answer to two decimal places.)

______ persons / year

1. Find all values of m so that the function y = e^mx is a solution of the given differential equation. (Enter your answers as a comma-separated list.)

y prime + 3y = 0

m =____

2. Determine whether Theorem 1.2.1 guarantees that the differential equation
y prime = square root y^2 -25 possesses a unique solution through the given point.
(1, 7)

A. Yes
B. No

Assume that in the absence of immigration and emigration, the growth of a country's population P(t) satisfies dP/dt = kP for some constant k > 0. Determine a differential equation governing the growing population P(t) of the country when individuals are allowed to immigrate into the country at a constant rate r > 0. (Use P for P(t).)

dP/dt = _____

What is the differential equation for the population P(t) of the country when individuals are allowed to emigrate at a constant rate r > 0?

dP/dt =______

In this problem, y = 1/(x2 + c) is a one-parameter family of solutions of the first-order
DE y' + 2xy2 = 0.Find a solution of the first-order IVP consisting of this differential equation and the given initial condition.

y(3) = 1/5

y = ___

Give the largest interval I over which the solution is defined.

A (0, infinity)
B. (2, infinity)
C. ( - infinity, infinity)
D. (0, 2)
E. (-2, infinity)

1. Determine a region of the xy-plane for which the given differential equation would have a unique solution whose graph passes through a point (x0, y0)in the region.

(4-y^2 ) y prime = x^2

A. A unique solution exists in the entire xy-plane.
B. A unique solution exists in the regions y < −2, −2 < y < 2, and y > 2.
C. A unique solution exists in the region consisting of all points in the xy-plane except (0, 2) and (0, −2).
D.A unique solution exists in the region y > −2.
E. A unique solution exists in the region y < 2.

What function do you know from calculus is such that its first derivative is itself? (Do not use the function f(x) = 0.)

f(x) = ___

The above function is a solution of which of the following differential equations?

A. y prime = y
B. y prime = y^2
C. y prime = 1
D. y prime = e^y
E. y prime = 2y

What function do you know from calculus is such that its first derivative is a constant multiple k of itself? (Do not use the function
f(x) = 0.)

f(x) = ___

The above function is a solution of which of the following differential equations?

A. y prime = k
B. y prime = y + k
C. y prime = e^ky
D. y prime = ky
E. y prime = y^k

1. In this problem, x = c1 cos t + c2 sin t is a two-parameter family of solutions of the second-order DE x'' + x = 0. Find a solution of the second-order IVP consisting of this differential equation and the given initial conditions.

x(pi/3) = square root of 3 / 2, x prime (pi/3) = 0

x = ____

d^2y/dx^2 + 3(dy/dx) + 2y = sin(e^x )