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9 Cards in this Set

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How many initial values will you need if you have a nth-order differential equations that you wish to fully solve?
you will need n initial value conditions, starting from

y(x) = y0, y'(x) = y1, y''(x) = y2, ..., all the way to y^(n-1)(x) = y(n-1)
What does the theory on the "Existence of a Unique Solution" say?
As long as the coefficients on the left hand side of the equation are all continuous. And the f(x) on the right hand side of the equation is continuous. If x = x0 is any point in this interval, then a solution y(x) of the initial-value problem (1) exist on the interval and is unique.
What is a boundary-value problem (BVP)?
A BVP is any linear equation of order two or greater in which the dependent variable y or its derivatives are specified at different points.

The different points specified are known as boundary conditions. Any solution to the differential equation must be a function that passes through these initial condition points.
What different type of solutions can a BVP have?
A BVP can have Many, One, or No Solutions.
What does homogenous mean? What does nonhomogenous mean?
Homogenous means that a function equals 0. As in g(x) = 0. Nonhomogenous means that a function equals something other than 0, as in f(x) = g(x), where g(x) is not zero.
What is the Quadratic Equation?
[-b (+|-) Sqrt[b^2 - 4(a)(c)] ]/2a, when an equation is of the form
am^2 + bm + c = 0
How do you solve a homogenous linear equation with constant coefficients?
Find the roots of the equation using whatever means neccesary.

The steps are:
i)make the assumption that the solution will contain r(t) = e^(mt), which mean r'(t) = m*e^(mt), etc.
ii) now make the equivilant substitutions into the equation.
iii) reorganize so that you have e^(mt)*G(x), find all solutions for G(x).
iiii) create solution of y = {(constant 1, 2, 3,...)*e^(all the solutions you got from part iii))}. so if you found 3 solutions in iii, you should have three parts to your y = something.
What is the function name to find roots for a polynomial equation in mathermatica? How high an order can it solve for?
Solve[3m^3 + 5m^2 + 10m == 0, m]

polynomial can be algebraicly solved if they are of degree less than 5
What is the mathematica equation for solving a differential equation?
DSolve[y''[x] + 2y'[x] +2y[x] == 0, y[x], x]