Podcast
Questions and Answers
What is the dependent variable in the differential equation 𝑑𝑃 = 𝑘𝑃 + 𝜔?
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?
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?
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?
What is the step size of the problem given by 𝑑𝑦 = 𝑡 − 𝑦², 𝑦(0) = 1, 0 ≤ 𝑡 ≤ 1, ∆𝑡 = 0.25?
Which of the following qualities refers to a differential equation having a unique solution?
Which of the following qualities refers to a differential equation having a unique solution?
What value of 𝑦 will NOT result in an equilibrium point of the differential equation 𝑑𝑦 = sin(𝑦)?
What value of 𝑦 will NOT result in an equilibrium point of the differential equation 𝑑𝑦 = sin(𝑦)?
What is the purpose of finding the equilibrium points in a differential equation?
What is the purpose of finding the equilibrium points in a differential equation?
What is the term for a differential equation that does not explicitly contain the independent variable?
What is the term for a differential equation that does not explicitly contain the independent variable?
What is the value of P for which the population would be constant in the logistic model?
What is the value of P for which the population would be constant in the logistic model?
What is the solution to the differential equation 𝑑𝑦 = 𝑦2−1?
What is the solution to the differential equation 𝑑𝑦 = 𝑦2−1?
How many solutions are there in the differential equation 𝑑𝑣𝑐 = − 𝑣𝑐?
How many solutions are there in the differential equation 𝑑𝑣𝑐 = − 𝑣𝑐?
What should be done to have an accurate approximate in Euler’s method?
What should be done to have an accurate approximate in Euler’s method?
What does it mean if all the solutions for the differential equation are the same and are within the same interval of time?
What does it mean if all the solutions for the differential equation are the same and are within the same interval of time?
Which of the following values of y will result to the equilibrium points of the differential equation 𝑑𝑦 = (𝑦 − 2)(𝑦 + 3)?
Which of the following values of y will result to the equilibrium points of the differential equation 𝑑𝑦 = (𝑦 − 2)(𝑦 + 3)?
What type of problem is the population model 𝑑𝑃 = 0.5𝑃(1 − 𝑃/200)?
What type of problem is the population model 𝑑𝑃 = 0.5𝑃(1 − 𝑃/200)?
What is the carrying capacity of the population in the model 𝑑𝑃 = 0.5𝑃(1 − 𝑃/200)?
What is the carrying capacity of the population in the model 𝑑𝑃 = 0.5𝑃(1 − 𝑃/200)?
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.
