Mathematics Field of Specialization Exam

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Questions and Answers

What is the dependent variable in the differential equation 𝑑𝑃 = 𝑘𝑃 + 𝜔?

𝑘

𝜔

𝑃 (correct)

𝑡

Which of the following is NOT an autonomous differential equation?

𝑑𝑥 = 1 + 𝑥² (correct)

𝑑𝑤 = 𝑤

𝑑𝑥 = 𝑒𝑥

𝑑𝑦 = 2 − 𝑦

What technique of solving differential equations involves using geometry to visualize the behavior of the model?

Euler's method

Numerical approach

Analytic approach

Qualitative approach (correct)

What is the step size of the problem given by 𝑑𝑦 = 𝑡 − 𝑦², 𝑦(0) = 1, 0 ≤ 𝑡 ≤ 1, ∆𝑡 = 0.25?

∆𝑡 = 0.25 Signup and view all the answers

Which of the following qualities refers to a differential equation having a unique solution?

Uniqueness Signup and view all the answers

What value of 𝑦 will NOT result in an equilibrium point of the differential equation 𝑑𝑦 = sin(𝑦)?

1 Signup and view all the answers

What is the purpose of finding the equilibrium points in a differential equation?

To determine the long-term behavior of the solution Signup and view all the answers

What is the term for a differential equation that does not explicitly contain the independent variable?

Autonomous differential equation Signup and view all the answers

What is the value of P for which the population would be constant in the logistic model?

P = N Signup and view all the answers

What is the solution to the differential equation 𝑑𝑦 = 𝑦2−1?

𝑦(𝑡) = 1 − 𝑡 Signup and view all the answers

How many solutions are there in the differential equation 𝑑𝑣𝑐 = − 𝑣𝑐?

1 Signup and view all the answers

What should be done to have an accurate approximate in Euler’s method?

Decrease the value of the step size Signup and view all the answers

What does it mean if all the solutions for the differential equation are the same and are within the same interval of time?

The solution to the differential equation is unique Signup and view all the answers

Which of the following values of y will result to the equilibrium points of the differential equation 𝑑𝑦 = (𝑦 − 2)(𝑦 + 3)?

I. -3 and II. 0 Signup and view all the answers

What type of problem is the population model 𝑑𝑃 = 0.5𝑃(1 − 𝑃/200)?

Logistic growth Signup and view all the answers

What is the carrying capacity of the population in the model 𝑑𝑃 = 0.5𝑃(1 − 𝑃/200)?

200 Signup and view all the answers

Study Notes

Differential Equations

Field of Specialization

Major in Mathematics
25 items

Instructions

1.5 hours to complete the test
Shade only one answer per question
Avoid erasures

Multiple Choice Questions

Dependent Variables

In the differential equation dP = kP + ω, the dependent variable is P

Autonomous Differential Equations

An autonomous differential equation does not involve time explicitly
Example: dy/dt = 2 - y
Example: dx/dt = e^x
NOT autonomous: dx/dt = 1 + x^2

Techniques of Solving Differential Equations

Qualitative approach: uses geometry to overview the behavior of the model
Analytic approach
Numerical approach: Euler's method
Euler's method: decreases the step size for accurate approximation

Existence and Uniqueness

Existence: the differential equation has a valid solution
Uniqueness: the differential equation has only one solution

Phase Lines

Equilibrium points: f(y) = 0
Finding equilibrium points in a differential equation
Example: dy/dt = sin(y)

Logistic Model

Population model: dP/dt = k(1 - P/N)P
Population is constant when P = N

Solution to a Differential Equation

Example: dy/dt = y^2 - 1
Solution: y(t) = 1 + t

Electrical Circuit

Differential equation: dv_c/dt = -v_c/RC
Initial condition: v_c(0) = 6
Slope field: number of solutions is 1

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Description

This 25-item exam is designed to test one's knowledge and skills in mathematics, with a time limit of 1.5 hours. It covers various topics in mathematics and is a major assessment for mathematics students.