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Questions and Answers
Which of the following equations represents a third-order differential equation?
Which of the following equations represents a third-order differential equation?
- y'' + 2y' + y = 0
- $\frac{d^3y}{dt^3} + 3 = 0$ (correct)
- f'(t) = -1
- y' = x - y
Newton's derivation of the height of a falling body involves several steps. Which of the following correctly outlines the progression of these steps?
Newton's derivation of the height of a falling body involves several steps. Which of the following correctly outlines the progression of these steps?
- Application of initial conditions → initial differential equation → integration to find velocity → integration to find position.
- Initial differential equation → integration to find velocity → integration to find position → application of initial conditions. (correct)
- Integration to find position → initial differential equation → integration to find velocity → application of initial conditions.
- Initial differential equation → integration to find position → integration to find velocity → application of initial conditions.
The rate at which the velocity of an object changes is 4 $m/s^2$. Which differential equation expresses this relationship?
The rate at which the velocity of an object changes is 4 $m/s^2$. Which differential equation expresses this relationship?
- $\frac{dv}{dt} = 4$ (correct)
- $\frac{d^2v}{dt^2} = 4$
- $\int v(t) dt = 4$
- $\frac{dv}{dt} = 4t$
The rate at which a drink warms or cools is proportional to the difference between the ambient temperature $T_a$ and the current temperature of the drink $T$. Which equation models this?
The rate at which a drink warms or cools is proportional to the difference between the ambient temperature $T_a$ and the current temperature of the drink $T$. Which equation models this?
If pressure $P$ applied to a gas increases at constant temperature, the rate of change of volume $V$ with respect to pressure is proportional to the reciprocal of the square of the pressure. Which differential equation expresses this?
If pressure $P$ applied to a gas increases at constant temperature, the rate of change of volume $V$ with respect to pressure is proportional to the reciprocal of the square of the pressure. Which differential equation expresses this?
Determine whether $y = -2x^5$ is a solution to the differential equation $\frac{dy}{dx} = \frac{y}{x}$.
Determine whether $y = -2x^5$ is a solution to the differential equation $\frac{dy}{dx} = \frac{y}{x}$.
Given the differential equation $y'' - 2y' - 8y = 0$, which of the following functions is a solution?
Given the differential equation $y'' - 2y' - 8y = 0$, which of the following functions is a solution?
A slope field visualizes solutions to differential equations. For the differential equation $\frac{dy}{dx} = x + y$, what is the slope at the coordinate point (1, 0)?
A slope field visualizes solutions to differential equations. For the differential equation $\frac{dy}{dx} = x + y$, what is the slope at the coordinate point (1, 0)?
A slope field for a certain differential equation is given. Which of the following could be the solution to the differential equation with the initial condition y(0) = 0?
A slope field for a certain differential equation is given. Which of the following could be the solution to the differential equation with the initial condition y(0) = 0?
Given a slope field, which statement about a solution $y = f(x)$ must be false?
Given a slope field, which statement about a solution $y = f(x)$ must be false?
Consider the differential equation $\frac{dy}{dx} = \frac{y + 1}{2}$. What steps would you take to find the particular solution passing through the point (1,0)?
Consider the differential equation $\frac{dy}{dx} = \frac{y + 1}{2}$. What steps would you take to find the particular solution passing through the point (1,0)?
Consider the given slope field. Which of the following could be a solution to the differential equation with the initial condition $y(0) = 1$?
Consider the given slope field. Which of the following could be a solution to the differential equation with the initial condition $y(0) = 1$?
Based on the slope field, which of the following could be a solution, $y = f(x)$, to the differential equation passing through the point (0, 0)?
Based on the slope field, which of the following could be a solution, $y = f(x)$, to the differential equation passing through the point (0, 0)?
For the differential equation $\frac{dy}{dx} = \frac{x^2}{y}$ for all $y \neq 0$, describe all points in the xy-plane for which $\frac{dy}{dx} = -2$.
For the differential equation $\frac{dy}{dx} = \frac{x^2}{y}$ for all $y \neq 0$, describe all points in the xy-plane for which $\frac{dy}{dx} = -2$.
Given a slope field for a certain differential equation, which of the following could the solution to the differential equation be?
Given a slope field for a certain differential equation, which of the following could the solution to the differential equation be?
Which of the following differential equations is separable?
Which of the following differential equations is separable?
What is the general solution to the differential equation $\frac{dy}{dx} = y \sin(x)$?
What is the general solution to the differential equation $\frac{dy}{dx} = y \sin(x)$?
To find a particular solution to a differential equation, what is the correct order of steps?
To find a particular solution to a differential equation, what is the correct order of steps?
What is the particular solution that satisfies the differential equation $\frac{dy}{dx} = \frac{x+1}{y}$ and the condition $y(0) = 1$?
What is the particular solution that satisfies the differential equation $\frac{dy}{dx} = \frac{x+1}{y}$ and the condition $y(0) = 1$?
The number of barrels of oil a company exports annually increases at a rate proportional to the number of barrels exported at that time. Initially, the company exports 5.4 million barrels, and after 6 years, it exports 10.8 million barrels. Which equation expresses number of barrels in terms of time?
The number of barrels of oil a company exports annually increases at a rate proportional to the number of barrels exported at that time. Initially, the company exports 5.4 million barrels, and after 6 years, it exports 10.8 million barrels. Which equation expresses number of barrels in terms of time?
A population grows at a rate proportional to its size, represented by $\frac{dP}{dt} = kP$. Which of the following demonstrates the steps to find the general solution?
A population grows at a rate proportional to its size, represented by $\frac{dP}{dt} = kP$. Which of the following demonstrates the steps to find the general solution?
Consider the differential equation $\frac{dy}{dx} = \frac{x - 2}{y}$. Let $y = f(x)$ be the particular solution satisfying the initial condition $f(1) = 2$. Find an equation of the line tangent to the graph of $y = f(x)$ at $x = 1$.
Consider the differential equation $\frac{dy}{dx} = \frac{x - 2}{y}$. Let $y = f(x)$ be the particular solution satisfying the initial condition $f(1) = 2$. Find an equation of the line tangent to the graph of $y = f(x)$ at $x = 1$.
Of the following, which are solutions to the differential equation $y'' + 4y = 0$?
Of the following, which are solutions to the differential equation $y'' + 4y = 0$?
Flashcards
What is a Differential Equation?
What is a Differential Equation?
An equation that involves an unknown function and its derivatives.
What defines the order of a Differential Equation?
What defines the order of a Differential Equation?
The order corresponds to the highest derivative in the differential equation.
What are solutions to differential equations?
What are solutions to differential equations?
Functions that satisfy the differential equation.
What is a Slope Field?
What is a Slope Field?
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What is a General Solution?
What is a General Solution?
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What is a Particular Solution?
What is a Particular Solution?
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What is Separation of Variables?
What is Separation of Variables?
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What is a Separable Differential Equation?
What is a Separable Differential Equation?
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What is Acceleration?
What is Acceleration?
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Newton's Law of Cooling/Warming
Newton's Law of Cooling/Warming
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Relationship between Pressure and Volume
Relationship between Pressure and Volume
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What is Exponential Growth?
What is Exponential Growth?
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Study Notes
- Unit 7 focuses on differential equations and their applications.
Learning Objectives
- Define differential equations.
- Understand and use slope fields.
- Find general solutions using separation of variables.
- Use initial conditions for particular solutions.
Differential Equations Defined
- A differential equation involves derivatives of a function.
- Solutions will be a function
Order of Differential Equations
- The order corresponds to the highest derivative in the equation.
- y'' + 2y' + 3 = 0 is a second-order differential equation.
- d³y/dt³ + 3 = 0 is a third-order differential equation.
- f'(t) = -1 is a first-order differential equation.
Solving Differential Equations
- Separable differential equations involve isolating variables on each side.
- Slope fields can visually represent solutions to more complex differential equations.
Newton's Derivation of Falling Body Equation
- Starts from d²h/dt² = -g, where h is height and g is gravitational acceleration.
- Integrating dv/dt = -g results in v = -gt + C, where C is a constant.
- Object's position h(t) integrates to -½gt² + Ct + D, with D as another constant.
- Constants C and D are determined by initial conditions: initial height (hâ‚€) and initial velocity (vâ‚€).
- Therefore, h(t) = -½gt² + v₀t + h₀.
Differential Equation Examples
- "The rate at which the velocity of an object is changing is 4 meters per second per second" can be represented as dv/dt = 4.
- The rate of drink cooling is proportional to the difference between ambient (Ta) and drink (T) temperatures (dT/dt = k(Ta - T)).
- Volume change rate is proportional to the inverse square of pressure can be written as dV/dP = k/P².
Verifying Solutions
- Substitute the proposed solution into the differential equation.
- Check if it satisfies the equation.
Slope Fields
- Graphically represent differential equations
- Tangent lines at (x, y) have a slope of x + y.
- Curves following tangent lines represent particular solutions
- These curves are specific solutions from point (2,-1) on a family of general solutions
Separable Differential Equations
- Can be written in the form g(y) dy = f(x) dx.
- Example: dy/dx = x/y becomes ∫ydy = ∫xdx.
- NOT Separable: dy/dx = x - y or dy/dx = e^(xy).
Finding General Solutions
- Solve the general solution of g'(x) = y sin(x).
- To find a general solution for dy/dx = (3x²+2) / (5 cos(y)), separation of variables is used.
Solving Explicitly for y
- Find a general solution to dy/dx = (cos(x)e^(sin(x))) / cos(y) and solve for y.
Solving Differential Equations
- Formula for all y >0 is dy/dx = y - y sin(x).
Particular Solutions
- Find the general solution first.
- Apply initial conditions to find the constant C.
- Solve for y explicitly if needed.
Example: Finding Particular Solutions
- Given dy/dx = (x+1) / y
- Find the general solution.
- Solve for y(0) = 1 using two methods: unsimplified and simplified.
Exponential Growth/Decay
- A population grows at a rate proportional to its size is dP/dt = kP (k is proportionality constant).
- The general solution is P(t) = P(0)e^(kt), where P(0) is the starting population.
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