You have been asked to prove or disprove the sentence For al

You have been asked to prove or disprove the sentence: For all integers m, if m^2 - 2m is even, then m is even. Evaluate the following proofs and disproofs and select all those that you agree are correct. [At least one is correct!] The negation of the claim states: if m is odd, then m^- 2m is odd. Taking m = 3, we see that m^2 - 2m = 3, also odd. Since this example agrees with the negation of the claim, it is a counterexample to the original statement, which must be false. Assume (hoping for a contradiction) that for some m m^2 - 2m is even and m is odd. Then m = 2k + 1 for some integer k, leading to m^2 - 2m = 4k^2 - 1; but this is an odd number, so our assumption was impossible and the theorem must be true Suppose m is a particular but arbitrarily chosen even number: then m = 2p for some integer p. This leads to the calculation m^2 - 2m = 4p^2 - 4p = 2(2p^2 - 2p). Since 2p^2 - 2p is an integer, this means m^2 - 2m is even, and the theorem is true. Assume (hoping for a contradiction) that if m^2 - 2m is even, then m is odd. But in the example m = 2, m^2 - 2m = 0, we see that m^2 - 2m is even but m is also even. So our assumption is disproved and the theorem must be true.

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We will now turn our attention to solving systems of simultaneous homogeneous first order linear differential equations. The solutions of such systems require much linear algebra (Math 220). But since it is not a prerequisite for this course, we have to limit ourselves to the simplest instances: those systems of two equations and two unknowns only. But first, we shall have a brief overview and learn some notations and terminology. A system of n linear first order differential equations in n unknowns (an n × n system of linear equations) has the general form: x1 = a11 x1 + a12 x2 + … + a1n xn + g1 x2 = a21 x1 + a22 x2 + … + a2n xn + g2 x3 = a31 x1 + a32 x2 + … + a3n xn + g3 (*) : : : : : : xn = an1 x1 + an2 x2 + … + ann xn + gn Where the coefficients aij’s, and gi ’s are arbitrary functions of t. If every term gi is constant zero, then the system is said to be homogeneous. Otherwise, it is a nonhomogeneous system if even one of the g’s is nonzero. © 2008, 2012 Zachary S Tseng D-1 - 2 The system (*) is most often given in a shorthand format as a matrix-vector equation, in the form: x = Ax + g + = n n nn n n n n n n g g g g x x x x a a a a a a a a a a a a x x x x : : : : ... : : : : : : : : ... ... ... : : 3 2 1 3 2 1 1 2 31 32 3 21 22 2 11 12 1 3 2 1 x A x g Where the matrix of coefficients, A, is called the coefficient matrix of the system. The vectors x, x, and g are x = n x x x x : 3 2 1 , x = n x x x x : 3 2 1 , g = n g g g g : 3 2 1 . For a homogeneous system, g is the zero vector. Hence it has the form x = Ax. © 2008, 2012 Zachary S Tseng D-1 - 3 Fact: Every n-th order linear equation is equivalent to a system of n first order linear equations. (This relation is not one-to-one. There are multiple systems thus associated with each linear equation, for n > 1.) Examples: (i) The mechanical vibration equation mu + u + k u = F(t) is equivalent to m F t x m x m k x x x ( ) 2 1 2 1 2 + = = Note that the system would be homogeneous (respectively, nonhomogeneous) if the original equation is homogeneous (respectively, nonhomogeneous). (ii) y 2y + 3y 4y = 0 is equivalent to x1 = x2 x2 = x3 x3 = 4 x1 3 x2 + 2 x3 © 2008, 2012 Zachary S Tseng D-1 - 4 This process can be easily generalized. Given an n-th order linear equation any (n) + an1 y (n1) + an2 y (n2) + … + a2 y + a1 y + a0 y = g(t). Make the substitutions: x1 = y, x2 = y, x3 = y, … , xn = y (n1), and xn = y (n) . The first n 1 equations follow thusly. Lastly, substitute the x’s into the original equation to rewrite it into the n-th equation and obtain the system of the form: x1 = x2 x2 = x3 x3 = x4 : : : : : : xn1 = xn n n n n n n n n a g t x a a x a a x a a x a a x ( ) ... 1 3 2 2 1 1 0 + = Note: The reverse is also true (mostly)* . Given an n × n system of linear equations, it can be rewritten into a single n-th order linear equation