MA1506 LECTURE NOTES CHAPTER 1 DIFFERENTIAL EQUATIONS 1.1 Introduction A differential equation is an equation that contains one or more derivatives of a differentiable function. [In this chapter we deal only with ordinary DEs‚ NOT partial DEs.] The order of a d.e. is the order of the equation’s highest order derivative; and a d.e. is linear if it can be put in the form any (n)(x)+an−1y (n−1)(x)+· · ·+a1y (1)(x)+a0y(x) = F‚ 1 where ai‚ 0 ≤ i ≤ n‚ and F are all functions of x. For example
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Unforced Response 2 The state equation for an unforced dynamic system is Assume the solution is x ( t ) = e At x ( 0 ) The derivative of eAt with respect to time is d ( e At ) dt Checking the solution x ( t ) = Ax ( t ) = Ae At x ( t ) = Ax ( t ) ⇒ Ae At x ( 0 ) = Ae At x ( 0 ) Letting Φ(t) = eAt‚ the solution is written as x (t ) = Φ (t ) x ( 0) The matrix Φ(t) is called the state transition matrix. The state transition matrix transforms the initial conditions
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The Beal Conjecture Background Mathematicians have long been intrigued by Pierre Fermat’s famous assertion that Ax + Bx = Cx is impossible (as stipulated) and the remark written in the margin of his book that he had a demonstration or "proof". This became known as Fermat’s Last Theorem (FLT) despite the lack of a proof. Andrew Wiles proved the relationship in 1994‚ though everyone agrees that Fermat’s proof could not possibly have been the proof discovered by Wiles. Number theorists remain divided
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Northwest logging photos’ analysis One photograph depicts a group of rural men. All of them are looking at the camera and some of them are seated and others stood on or near a huge tree’s trunk. Few of them hold on their hands work equipment such as axs which help them to cut down and dismember tree’s trunk. Thus‚ this single picture illustrate theme of work shown by piece of wood around and work equipments. Another element that emerges from this picture is the exploitation of wood as a daily activity
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citizens. The killer attacked people in their sleep‚ using weapons only from the victims households‚ primarily an ax. The first attack happened on May 2‚ 1918. Victims Catherine and Joseph Maggi had been attacked by an ax‚ and also had their throats slit. Joseph’s brothers discovered the two‚ and claim nothing was heard or seen. No valuables were taken‚ only discovered a bloody ax by the door. Second attack occurred roughly a month later‚ on the 28. Louis Besumer and Anna Lowe were discovered
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system: x + 2y − 3z = a 2x + 3y + 3z = b 5x + 9y − 6z = c is consistent for any values of a‚b and c that satisfy that equation. 7. Show that if u and v are solution to the linear system Ax = b‚ then u − v is a solution to the associated homogeneous system Ax = 0. 8. If A is an n × n matrix‚ the homogeneous system Ax = 0 has a nontrivial solution if and only if A is singular. 9. Find all values of a for which the inverse of 1 1 0 A= 1 0 0 1 2 a exists. What is A−1 ? 10. For what values
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GRAVITATION Kepler’s Laws Towards the end of the sixteenth century‚ Tycho Brahe collected a huge amount of data giving precise measurements of the position of planets. Johannes Kepler‚ after a detailed analysis of the measurements announced three laws in 1619. 1. The orbit of each planet is an ellipse which has the Sun at one of its foci. 2. Each planet moves in such a way that the (imaginary) line joining it to the Sun sweeps out equal areas in equal times. 3. The squares of the periods
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Review of Algebra 2 s REVIEW OF ALGEBRA Review of Algebra q q q q q q q q q q q q q q q Here we review the basic rules and procedures of algebra that you need to know in order to be successful in calculus. Arithmetic Operations The real numbers have the following properties: a b b a ab a b c a b ab c ab ac In particular‚ putting a b and so b c b c ba c (Commutative Law) (Associative Law) (Distributive law) ab c a bc 1 in the
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yn-1.dydx ‚where y=f(x) * dydx=1dxdy * ddxsinx=cosx * ddxcscx=-cscx.cotx * ddxcosx=-sinx * ddxsecx=secx.tanx * ddxtanx=sec2x * ddxcotx=-csc2x * xn dx=xn+1n+1+c‚ where n≠-1 * x-1 dx=lnx+c * (ax+b)n dx= 1a(ax+b)n adx= 1a×(ax+b)n+1n+1+c * sinx dx= -cosx+c * cosx dx= sinx+c * cosec2x dx= -cotx+c * sec2x
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function is a relation in which each element of the domain is paired with exactly one element in the range. Two types of functions are the exponential functions and the logarithmic functions. Exponential functions are the functions in the form of y = ax‚ where ’’a’’ is a positive real number‚ greater than zero and not equal to one. Logarithmic functions are the inverse of exponential functions‚ y = loga x‚ where ’’a’’ is greater to zero and not equal to one. These functions have certain differences
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