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12 Cards in this Set
- Front
- Back
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What is a Differential Equation (DE)?
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A differential Equation is any equation containing the derivatives of one or more dependent variables with respect to one or more independent variables.
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What are the three terms used to categorize differential equations?
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Classification by Type.
Classification by order. Classification by linearity. |
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What does it mean for a differential equation to be classified by type. What are the types?
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If an equation contains only ordinary derivatives of one or more dependent variables with respect to a single independent variable, it is said to be an ordinary differential equation (ODE).
An equation containing the partial derivative of one or more dependent variables of two or more independent variables, it is said to be a partial differential equation (PDE). |
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What is Leibniz notation?
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Leibniz notation is a derivative notation of the form dy/dx, d^2y/dx^2, ..., d^ny/dx^n
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What is Prime Notation?
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Prime Notation is a derivative notation of the form y', y'', y''', y^(4), y^(n). Not that you only continue using the prime (y') notation up to the third derivation. After that, you should use a number, or use Liebniz.
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What is Newtons dot notation?
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Newtoons dot notation is a notation were you simply place as many dots above the variable as are neccesary to indicate the number of derivations.
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What does it mean for a differential equation to be classified by Order?
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The order of a differential equations (either ODE or PDE) is the order of the highest derivative in the equation.
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What does it mean for a differential equation to be classified by Linearity?
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An nth order differntial equation is said to be linear if F is linear in y, y', ..., y(n)
nonlinear ordinary differential equations are simply differential equations that are not linear. |
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What is a solution to a differntial equation?
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Any function defined on an interval M and possessing at least n derivates that are continuous on M, which when substituted into an nth-order ordinary differential equation reduces the equation to an identity, is said to be a solution of the equation on the identity.
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What does it mean to have a interval of definition for a a differential equation?
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The interval of definition, also known as the interval of existence, the interval of validity, or the domain is the interval on which the original funciton and all derivative funtions are valid. It can be an open interval (a, b), a closed interval [a, b], an infinit interval (a , infinity) , and so on.
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Reagarding differntial equations, how is a linear first order differential equation modeled?
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A first-order differential equation of the form
P(x)*(dy/dx) + Q(x)*y = G(x) is said to be a linear equation. |
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How do you solve any linear first order Differential Equation of the form:
(dy/dx) + P(x)y = G(x) |
i) Put a linear equation into standard form, standard form is
(dy/dx) + P(x)y = G(x) ii) From the standard form, identify P(x) and then find the integrating factor e^(Integral[P(x)dx]) iii) Multiply the standard form of the equation by the integrating factor. The left-hand side of the the resulting equation is automatically the derivative of the integrating factor and y: (d/dx)[e^(Integral[P(x)dx])y] = e^(Integral[P(x)dx])G(x) iiii) Integrate both sides of this last equation |
