Exam 10: Introduction to Differential Equations

Thirty birds were brought to a zoo 10 years ago. At present, there are 60 birds in the zoo. The zoo can support a maximum of 120 birds. Assuming a logistic growth model, when will the bird population reach 80, 100, and 120 birds?

Solve the initial value problem .

Consider the circuit shown in the following figure. At the switch is closed and the current passes through the circuit. The constant voltage is the sum of the voltage across the resistor and the voltage across the inductor. A) Set up an initial value problem satisfied by . B) Solve for the current . C) What is the current after a long time?

Consider the initial value problem . A) Use Euler's Method with to approximate B) Use Euler's Method with to approximate C) Solve the initial value problem and estimate the errors in parts A and B.

Let be the population of a certain animal species, satisfying the logistic equation A) Determine the equilibrium solutions and their stabilities. B) What is the long term behavior of the population? C) Find the value of at the inflection point and explain its meaning referring to the population growth. D) Solve the logistic equation and verify your answer to B.

A rat population for a certain field is 300 at the beginning of the year 2002.. After 4 years, the population increased to 700. Assuming logistic growth with a carrying capacity of 900, express the population as a function of t, the number of years since the beginning of 2002.

A tank in the shape of a prism and filled with water has a height of 5 m. The cross sections are equilateral triangles with a side of 1 m. Water leaks through a hole of area 0.01 at the bottom of the tank. A) Find the water level at time . B) How long does it take for the tank to empty?

Use Euler's method with step size to approximate where is the solution to the initial value problem .

Solve the initial value problems. A) B)

Use Euler's method with to approximate where is the solution of . Give your answer to five decimal places.

Use Euler's method with step size to approximate where is the solution to the initial value problem .

Find all the curves with the following property: the y-intercept of the tangent line at any point on the curve is equal to .

Consider the initial value problem . A) Use Euler's method with to approximate the solution at the point . B) Solve the initial value problem and find the exact solution.

A conical tank filled with water has a height of 7 m and a top radius of 2 m. Water leaks through a hole of area at the bottom. A) Find the water level at time . B) How long does it takes for the tank to empty?

A deer population for a certain area was 400 at the beginning of the year 1995. After 2 years, the population increased to 500. Assuming logistic growth with a carrying capacity of 800, express the population as a function of t, the number of years since the beginning of 1995.

Solve the initial value problem .

Solve the differential equations. A) B) . (Hint: factor the denominator.)

A rat population in a certain field is initially 500. After 3 years, the population increases to 700. Assuming logistic growth with a carrying capacity of 1000, how long after reaching 700 will it take for the population to reach 800?

Solve the initial value problem .

Use Euler's method with to approximate , where is the solution of the initial value problem .