Differential Equations
Definition
Differential equations are mathematical equations that involve derivatives. They describe how a function changes over time or in relation to other variables.
Analogy
Think of differential equations as road maps that show the changing terrain as you drive along a winding road. The equation tells you how the landscape (function) evolves as you move forward.
Related terms
Derivatives: Derivatives are rates of change, representing how a function's output changes with respect to its input.
Function: A function is a rule that assigns each input value to exactly one output value.
Integral: Integrals are the reverse operation of derivatives, representing the accumulation or total amount of change in a function over an interval.
"Differential Equations" appears in:
Study guides (2)
Additional resources (4)
Practice Questions (20+)
- The rate of disappearance of a drug in a patient's bloodstream is directly proportional to the concentration of the drug at a given time. If C(t) represents the drug concentration at time t, which of the following differential equations can represent this situation?
- As a snowball melts, the rate of change of its mass is inversely proportional to the square of its mass at a given time. If m(t) is the mass of the snowball at time t, which of the following differential equations can represent this situation?
- The population of a city is growing at a rate proportional to both its current population and the difference between its current population and a carrying capacity, K. If P(t) represents the population at time t, which of the following differential equations represents this situation?
- t seconds after a Calculus textbook is dropped from a building, the rate of change of the textbook’s position is proportional to the amount of time passed. If y(t) is the position of the textbook at time t, which of the following differential equations represents this situation?
- The rate of change of the height h of a tree with respect to time t is directly proportional to the natural logarithm of h. Which of the following differential equations represents this situation, where k is a proportionality constant?
- The rate of change of a quantity A is directly proportional to the square of A and inversely proportional to the time t. Which of the following differential equations represents this situation?
- The kinetic energy E of a satellite decreases with respect to time t proportional to the square of the satellite’s velocity, v, and inversely proportional to the satellite’s mass, m. Which of the following differential equations represents this situation?
- A scientist observes that, when the temperature T of a substance is doubled, the rate of change in its mass m with respect to the time t is also doubled. Which of the following differential equations could represent the situation?
- The rate of change of a parachuter’s speed is the sum of two terms: a constant gravity that speeds up the parachuter, and air resistance that slows down the parachuter, which is proportional to the parachuter’s speed. Which of the following differential equations represents this situation?
- The second derivative of the human population P with respect to time is proportional to the first derivative of P with respect to time. Which of the following differential equations represents the situation?
- Which subject incorporates the process of verifying solutions to differential equations in the study of population models?
- Which common mistake should be avoided when verifying solutions for differential equations?
- In which subject is the process of verifying solutions to differential equations commonly applied in the study of motion?
- What is the purpose of substituting values for x when verifying solutions to differential equations?
- Which field of study utilizes the process of verifying solutions to differential equations to predict the future state of systems?
- How are slope fields useful in solving differential equations?
- What is the purpose of using a calculator in analyzing differential equations?
- How does finding solutions to differential equations contribute to predicting financial trends?
- How can ignoring domain restrictions when solving differential equations affect the validity of the solutions?
- Which of the following differential equations represents exponential decay?
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