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Questions and Answers
The differential equation $(e^y + 1)\cos(x) dx + e^y \sin(x) dy = 0$ is:
The differential equation $(e^y + 1)\cos(x) dx + e^y \sin(x) dy = 0$ is:
- Linear but not separable
- Separable but not linear (correct)
- Linear and separable
- Neither linear nor separable
Given the differential equation $x^2y dx - (x^3 + y^3) dy = 0$, what is the appropriate substitution to make it homogeneous?
Given the differential equation $x^2y dx - (x^3 + y^3) dy = 0$, what is the appropriate substitution to make it homogeneous?
- v = x + y
- v = x/y
- v = y/x (correct)
- v = xy
The differential equation $(D^2 - 3D + 2)y = 0$ has a general solution of the form $y = c_1e^{x} + c_2e^{2x}$. Given this, what are the roots of the auxiliary equation?
The differential equation $(D^2 - 3D + 2)y = 0$ has a general solution of the form $y = c_1e^{x} + c_2e^{2x}$. Given this, what are the roots of the auxiliary equation?
- -1, -2
- -1, 2
- 1, 3
- 1, 2 (correct)
For the differential equation $(D^2 + 1)y = \sin(2x) \sin(x)$, which trigonometric identity would be most helpful in simplifying the right-hand side before finding a particular solution?
For the differential equation $(D^2 + 1)y = \sin(2x) \sin(x)$, which trigonometric identity would be most helpful in simplifying the right-hand side before finding a particular solution?
Given the equation $(D^2 + 4)y = \sin(2x)$, what form should the particular solution $y_p$ take when using the method of undetermined coefficients?
Given the equation $(D^2 + 4)y = \sin(2x)$, what form should the particular solution $y_p$ take when using the method of undetermined coefficients?
Consider the differential equation $(D^2 + 3D + 2)y = x^2$. What form should the particular solution $y_p$ take when using the method of undetermined coefficients?
Consider the differential equation $(D^2 + 3D + 2)y = x^2$. What form should the particular solution $y_p$ take when using the method of undetermined coefficients?
For the initial value problem $y'' + y = 8e^{-2t}\sin(t)$, with $y(0) = 0$ and $y'(0) = 0$, what method would be most appropriate to find the solution?
For the initial value problem $y'' + y = 8e^{-2t}\sin(t)$, with $y(0) = 0$ and $y'(0) = 0$, what method would be most appropriate to find the solution?
Given the differential equation $(D^2 - 2D + 5)y = e^{2x}\sin(x)$, what is the correct form of the particular solution using the method of undetermined coefficients?
Given the differential equation $(D^2 - 2D + 5)y = e^{2x}\sin(x)$, what is the correct form of the particular solution using the method of undetermined coefficients?
What is the order and degree of the following ordinary differential equation (ODE)?
$(\frac{d^3y}{dx^3})^{\frac{2}{3}} = 1 + 2(\frac{dy}{dx}) \cdot \frac{d^2y}{dx^2}$
What is the order and degree of the following ordinary differential equation (ODE)?
$(\frac{d^3y}{dx^3})^{\frac{2}{3}} = 1 + 2(\frac{dy}{dx}) \cdot \frac{d^2y}{dx^2}$
Which differential equation (DE) represents the family of curves $y = a e^{2x} - b e^{-2x}$, where $a$ and $b$ are arbitrary parameters?
Which differential equation (DE) represents the family of curves $y = a e^{2x} - b e^{-2x}$, where $a$ and $b$ are arbitrary parameters?
Which differential equation represents all circles of radius 'a' in the x-y plane?
Which differential equation represents all circles of radius 'a' in the x-y plane?
Given the separable differential equation $\sec^2(x) \tan(y) dx + \sec^2(y) \tan(x) dy = 0$, what is the general solution?
Given the separable differential equation $\sec^2(x) \tan(y) dx + \sec^2(y) \tan(x) dy = 0$, what is the general solution?
What is a suitable substitution to solve the following differential equation? $\frac{dy}{dx} = e^{x-y} + x^2 e^{-y}$
What is a suitable substitution to solve the following differential equation? $\frac{dy}{dx} = e^{x-y} + x^2 e^{-y}$
Consider the homogeneous differential equation $(x^2 - 3y^2)dx + 2xy dy = 0$. What substitution would be most appropriate to solve this equation?
Consider the homogeneous differential equation $(x^2 - 3y^2)dx + 2xy dy = 0$. What substitution would be most appropriate to solve this equation?
Solve the following first-order linear differential equation: $(1 + x^2)\frac{dy}{dx} + 2xy = 4x^2$.
Solve the following first-order linear differential equation: $(1 + x^2)\frac{dy}{dx} + 2xy = 4x^2$.
Given the differential equation $\frac{dy}{dx} + y \cos(x) = 2 \sin^2(x)$, what type of equation is it and how would you solve for it?
Given the differential equation $\frac{dy}{dx} + y \cos(x) = 2 \sin^2(x)$, what type of equation is it and how would you solve for it?
For the differential equation $(1 + y^2)dx + (x - e^{-\arctan(y)})dy = 0$, what integrating factor would help to solve it?
For the differential equation $(1 + y^2)dx + (x - e^{-\arctan(y)})dy = 0$, what integrating factor would help to solve it?
Consider the differential equation $\frac{dy}{dx} = x^3y^3 - xy$. What type of equation is this and how would you proceed to solve it?
Consider the differential equation $\frac{dy}{dx} = x^3y^3 - xy$. What type of equation is this and how would you proceed to solve it?
Flashcards
Differential Equation
Differential Equation
An equation involving derivatives of a function with respect to one or more independent variables.
General Solution
General Solution
A solution that contains arbitrary constants; represents a family of solutions.
Particular Solution
Particular Solution
A solution obtained from the general solution by assigning specific values to the arbitrary constants.
Homogeneous Linear DE
Homogeneous Linear DE
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Solving a DE
Solving a DE
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Homogeneous Equation (DE)
Homogeneous Equation (DE)
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Exact Differential Equation
Exact Differential Equation
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General Solution (Linear ODE)
General Solution (Linear ODE)
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Order of an ODE
Order of an ODE
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Degree of an ODE
Degree of an ODE
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Differential Equation (DE)
Differential Equation (DE)
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Separable DE
Separable DE
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DE Reducible to Separable Form
DE Reducible to Separable Form
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Homogeneous Function
Homogeneous Function
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Homogeneous DE
Homogeneous DE
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Linear DE
Linear DE
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Integrating Factor (IF)
Integrating Factor (IF)
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Bernoulli's Equation
Bernoulli's Equation
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Study Notes
- Ordinary Differential Equations (ODEs) problems involve finding the order and degree
- Requires determining the differential equation (DE) of a curve given its equation, which includes parameters that need eliminating
- Finding DEs representing geometric properties, such as circles with a specific radius in the x-y plane challenges students to connect geometric concepts with differential equations
- Several problems focus on solving first-order ODEs, including those involving trigonometric functions, exponential functions, and various algebraic manipulations
- Includes finding a relation between x and y given a differential coefficient equal to a function of x and y, emphasizing the relationship between variables without explicit derivatives
- Many problems focus on solving second-order and third-order linear ODEs
- Requires students to find general solutions for equations with constant coefficients
- Some equations are homogeneous or non-homogeneous
- Includes initial value problems, where initial conditions are provided to find a particular solution
Solving Techniques for ODEs
- Problems may require using integrating factors, separation of variables, or recognizing exact equations
- Solving ODEs can be approached by undetermined coefficients, variation of parameters, or using the characteristic equation for homogeneous equations
- Several problems require solving ODEs with trigonometric functions
- Techniques include using trigonometric identities, reduction of order, or specific methods for handling trigonometric forcing functions
- Solving ODEs with exponential functions often involves finding particular solutions that account for the exponential term
- Includes handling cases where the exponential term is part of the homogeneous solution
- Involves solving ODEs with polynomial terms
- Emphasizes using power series solutions or other appropriate methods for polynomial-related ODEs
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Description
Practice problems for solving ordinary differential equations (ODEs). Includes identifying order and degree, forming differential equations from curves, and solving first, second, and third-order linear ODEs. Covers homogeneous and non-homogeneous equations with constant coefficients and initial value problems.
