Differential Equations MATH 225
Colorado School of Mines
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This 3 page Class Notes was uploaded by Diana Prosacco on Monday October 5, 2015. The Class Notes belongs to MATH 225 at Colorado School of Mines taught by Staff in Fall. Since its upload, it has received 16 views. For similar materials see /class/219616/math-225-colorado-school-of-mines in Mathematics (M) at Colorado School of Mines.
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Date Created: 10/05/15
15 Existence and Uniqueness Theorems Existence Theorem Suppose fty is a continuous function in a rectangle of the form tya lt t lt bc lt y lt d in the ty plane If t0y0 is a point in this rectangle then there exists an e gt O and a function yt defined for to e lt t lt to e that solves the initial value piroblem y dt ay Mo 90 8 Uniqueness Theorem Suppose fty and a f are 9 continuous functions in a rectangle of the form tya lt t lt bc lt y lt d in the ty plane If t0y0 is a point in this rectangle and if y1t and y2t are two functions that solve the initial value problem 2 ay y to yo for all t in the interval to e lt t lt to 6 6 is a positive number then 31105 31205 for to e lt t lt to e That is the solution to the initial value problem is unique Free Fall Linear Friction Suppose we have a mass7 m7 which is permitted to freefall in the atmosphere near the averagesurface of the Earth Assume that the dissipative force of friction is assumed to be proportional to the velocity7 v of the mass and from Newton7s laws derive a differential equation modeling the velocity of the mass as a function of time Solve this differential equation assuming the following initial conditions and describe the longterm asymptotic of the dynamics of the mass7 mg lly0lti7 7 mg 2iy077 7 mg Sly0gti7 7 where g is the gravitational constant of the universe and 7 is the coefficient of kinetic frictionl Assuming that m g 7 1 solutions to various initial value problems and comment on the results Free Fall Quadratic Friction It has been argued that for certain geometries dissipative forces are proportional to the square of the velocity You might want to think about this in the same sense that Taylor7s theorem argues that if a curve is locally concave then a quadratic function would give a better local approximation than a linear function That is7 the relation between force and velocity need not be as simple as a linear one Under these new assumptions derive from Newton7s laws a mathematical model that models the velocity of the mass as a function of time Solve this differential equation assuming the following initial conditions and describe the longterm asymptotic of the dynamics of the mass7 1 gm lt my 2 y0 V 7 3i y0 gt E where g is the gravitational constant of the universe and 7 is the coefficient of kinetic frictionl Assuming that m g 7 1 solutions to various initial value problems and comment on the results Hyperbolic Trigonometric Functions We will see later in this class that the exponential function is fundamentally related to the sine and cosine functions This can be seen through the use of Taylor s theorem and the complex number system it is because of this relationship that exponential function can be made to behave similarly to trigonometric functions in terms of differential and integral calculus These functions casually come up in practice to compactly write solutions to common physical problems in response to this I have constructed the following list of de nitions and properties which may or may not be useful We begin with the following de nitions for hyperbolic cosine and hyperbolic sine1 coshz g 6 671 l sinhz g 67 7 6 7 2 From l2 the following can be derived 1 Symmetry Properties sinh7z 7 sinhz 3 cosh7z coshz 4 2 De nition of Hyperbolic Tangent 7 sinhz tanhz 7 m 5 3 Some Hyperbolic Trigonometric ldentities2 coshz2 7 sinhz2 l 6 sechz2 tanhz2 l 7 4 Rules of Differentiation d coshz T s1nhz 8 d sinhz T coshz 9 W sechx2 10 5 Standard lntegrals IQLW arcsinh C 11 arccosh C 12 a2iixx2 iarctanh C 12 lt a2 13 1Wikipedia httpenwikipedia orgwikiHyperbolicifunction has the following useful overview of these functions Just as the points x y cost sint t 6 R rm a circle with a unit radius the points x y cosht sinht t 6 R form the right half of the equilateral hyperbola Hyperbolic functions are also useful because they occur in the solutions of some important linear differential equations notably that de ning the shape of a hanging cable the catenary and Laplace7s equation in Cartesian coordinates which is important in many areas of physics including electromagnetic theory heat transfer uid dynamics and special relativity A more complete listing may be found here httpquotWquot 1 L L I L ichtm1
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