A differential equation is linear if it can be expressed as a linear combination of the unknown function and its derivatives. For example, y′′+2xy′−3y=0y′′+2xy′−3y=0 is linear because each term involves y, y′, or y′′ raised to the power of 1. CANT BE SQUARED OR CUBED Is it possible to say something about the solutions to a differential equation without actually solving it? If so, what can be said? Explicit Solution What does it mean for a differential equation to be linear? Critical Points

What does it mean for a differential equation to be linear?

A differential equation is separable if it can be written in the form g(y)dy=h(x)dx, where g(y) depends only on y and h(x) depends only on x. dy/dx = xy can be written as dy/y = x dx exact differential equation Fundamental set of solutions What does it mean for a differential equation to be separable? Non-homogenous Differential Equation:

What does it mean for a differential equation to be separable?

An ordinary differential equation (ODE) involves only ordinary derivatives with respect to a single independent variable. It does not include partial derivatives. ar^2+br+c=0 What is this? What does it mean for a differential equation to be ordinary? Uniqueness of Solution Explicit Solution

What does it mean for a differential equation to be ordinary?

The highest order derivative in the expression Homogenous Differential Equation What does it mean for a differential equation to be separable? ar^2+br+c=0 What is this? What is the order of a differential equation?

What is the order of a differential equation?

Yes, it is possible. For example, one can determine whether solutions are bounded or unbounded, whether they approach certain values or behaviors as �x approaches infinity, etc. Is it possible to say something about the solutions to a differential equation without actually solving it? If so, what can be said? What's another name for the euler method? differential form What is an initial value problem?

Is it possible to say something about the solutions to a differential equation without actually solving it? If so, what can be said?

Substitute the function into the differential equation and verify that the equation holds true for all values of the independent variable within the domain of the function. What's another name for the euler method? Threshold Level Method of Undetermined Coefficients How does one know if a certain function is a solution to a given differential equation?

How does one know if a certain function is a solution to a given differential equation?

Yes, a general solution provides a framework that encompasses all possible solutions to the differential equation. What is an initial value problem? Characteristic Equation Linearization Does it always make sense to consider a general solution to a differential equation?

Does it always make sense to consider a general solution to a differential equation?

It is a problem in which we are given an initial value, a value of a function that we know beforehand. Linearization Is it possible to say something about the solutions to a differential equation without actually solving it? If so, what can be said? What is an initial value problem? integrating factor

What is an initial value problem?

Differential EQ Study Set-Exam One

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Calculus - Geometry

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A differential equation is linear if it can be expressed as a linear combination of the unknown function and its derivatives. For example, y′′+2xy′−3y=0y′′+2xy′−3y=0 is linear because each term involves y, y′, or y′′ raised to the power of 1. CANT BE SQUARED OR CUBED

A differential equation is separable if it can be written in the form g(y)dy=h(x)dx, where g(y) depends only on y and h(x) depends only on x. dy/dx = xy can be written as dy/y = x dx

An ordinary differential equation (ODE) involves only ordinary derivatives with respect to a single independent variable. It does not include partial derivatives.

The highest order derivative in the expression

Yes, it is possible. For example, one can determine whether solutions are bounded or unbounded, whether they approach certain values or behaviors as �x approaches infinity, etc.

Substitute the function into the differential equation and verify that the equation holds true for all values of the independent variable within the domain of the function.

Yes, a general solution provides a framework that encompasses all possible solutions to the differential equation.

It is a problem in which we are given an initial value, a value of a function that we know beforehand.

An initial value problem aims to find a particular solution that satisfies both the differential equation and the given initial conditions. This particular solution is often derived from the general solution by applying the initial conditions.

The domain of a solution to a DE depends on the domain of the equation itself and any additional constraints imposed by the initial conditions or boundary conditions. Avoiding singularities or discontinuities.

Involve only derivatives of y and terms involving y, and they're set to 0 with no other independent variable present. General homogeneous equations are valid in any system of units. A differential equation of the form f(x,y)dy = g(x,y)dx is said to be homogeneous differential equation if the degree of f(x,y) and g(x, y) is same

A non-homogenous differential equation includes terms that are not proportional to the dependent variable or its derivatives.

An equation involving partial derivatives of one or more unknown functions of two or more independent variables

A differential equation containing one or more functions of one independent variable and the derivatives of those functions.

the process of taking the gradient of a nonlinear function with respect to all variables and creating a linear representation at that point. Approximating a nonlinear eq by a linear one.

Uniqueness deals with having only one solution to an initial value problem. Nonunique- ness deals with more than one solution. A differential equation is said to have a unique solution if, for each initial or boundary condition, there is exactly one solution that satisfies the equation and the given condition.

e^(∫P(x) dx)

M(x,y)dx + N(x,y)dy = 0

Implicit solution means a solution in which dependent variable is not separated and explicit means dependent variable is separated.

a solution in which the dependent variable is expressed solely in terms of the independent variable and constants

They correspond to no change or variation in the value of y as t increases.

a point on a graph where the derivative is either zero or undefined.

level below which there are no visible or noticeable effects

we conclude that a necessary condition for exactness is that My=Nx

is an algorithm for approximating the solution to an initial value problem by following the tangent lines while we take horizontal steps across the t-axis

a method for converting any linear differential equation to another linear differential equation of lower order, and then constructing the general solution to the original differential equation using the general solution to the lower-order equation

A method for finding a particular solution to some non-homogeneous 2nd order linear equations by guessing a form based on the form of the forcing function. Know the rules for forming the guess

another way to solve nonhomogeneous linear differential equations, be they second order, ay′′ + by′ + cy = g

r^2 + pr + q = 0

{y₁,y₂} is a fundamental set of solutions to (H) provided: 1. y₁ and y₂ both solve (H) 2. y₁ and y₂ are linearly independent

Characteristic Equation

Tangent Line Method

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Differential EQ Study Set-Exam One

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