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UNIVERSITY BILKENT Department of Mathematics MATH 225, LINEAR ALGEBRA and DIFFERENTIAL EQUATIONS Homework set # 16 U.Muan g July 16, 2008
FUNDAMENTAL SET OF SOLUTIONS 1) a) b) c) Find the Wronskian of the following given pair of functions: e2x , e-3x/2 . x, xex . ex sin x, ex cos x.
2) Determine the largest interval in which the given I.V.P. is certain to have a unique twice differentiable solution. Do not find the solution. a) (x - 1)y - 3xy + 4y = sin x, y(-2) = 2, y (-2) = 1. b) y + (cos x)y + 3(ln |x|)y = 0, y(2) = 3, y (2) = 1. c) (x - 2)y + y + (x - 2)(tan x)y = 0, y(3) = 1, y (3) = 2. 3) Verify that y1 (x) = 1 and y2 (x) = x1/2 are two L.I. solutions of yy + (y )2 = 0, x > 0.
Then show that c1 + c2 x1/2 is not, in general a solution of the equation. Why not? 4) If Wronskian W of f and g is x2 ex and f (x) = x, find g(x).
5) Verify that the functions y1 and y2 are solutions of the following given equations. Do they constitute a fundamental set of solutions. a) y + 4y = 0, y1 (x) = cos 2x, y2 (x) = sin 2x. 2 b) x y - x(x + 2)y + (x + 2)y = 0, x > 0, y1 (x) = x, y2 (x) = xex . HOMOGENOUS, CONSTANT COEFFICIENT D.E. 6) a) b) c) Find the general solution of the D.E. following y + 5y = 0. y - 9y + 9y = 0. 2y - 3y + y = 0.
7) Find the solution of the following I.V.P. a) y + 4y + 3y = 0, y(0) = 2, y (0) = -1. b) y + 8y - 9y = 0, y(1) = 1, y (1) = 0. 8) Find so that the solution of the I.V.P. y - y - 2y = 0, y(0) = , y (0) = 2.
approaches to zero as x . COMPLEX ROOTS OF THE C.E. 9) a) b) c) Find the general solution of the following D.E. y - 2y + 2y = 0. 9y + 9y - 4y = 0. 4y + 9y = 0.
10) Find the general solution of the following I.V.P. and sketch the graph of the solution describe its behavior for increasing x. a) y + 4y + 5y = 0, y(0) = 1, y (0) = 0. b) y + 2y + 2y = 0, y(/4) = 2, y (/4) = -2. 11) If it is possible transform the following D.E's into one with constant coefficients and then find the general solution. 2 a) y + xy + e-x y = 0, - < x < . 2 3 b) xy + (x - 1)y + x y = 0, 0 < x < . VARIABLE COEFFICIENT D.E., EULER'S EQUATION 12) Find the transformation z = u(x) to transform the Euler's equation x2 y + xy + y = 0, x>0
where , are real constants, into an equation with constant coefficients. Use the above result to solve the following D.E. for x > 0. a) x2 y + xy + y = 0. b) x2 y - 4xy - 6y = 0.

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University of Michigan-Dearborn - CHEM - 225

University of Michigan-Dearborn - CHEM - 225

University of Michigan-Dearborn - CHEM - 225

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University of Michigan-Dearborn - CHEM - 225

University of Michigan-Dearborn - CHEM - 225

University of Michigan-Dearborn - CHEM - 225

CIIEMISTRYTEST #2 MAY 29,2009225NAMEGOOD LUCK!Atomicnumbers:3. 4._(23)(13)H: 1, C:6,N:7, O:8, S = 16, Cl:17, Br:35 TOTAL(75)l. (12 Points) STRUCTURALISOMERISM.a.(-/6)Considerthecyclichydrocarbon(compoundA)below.CHzCHe CHzCHscompound Acycl

University of Michigan-Dearborn - CHEM - 225

University of Michigan-Dearborn - CHEM - 225

University of Michigan-Dearborn - CHEM - 225

University of Michigan-Dearborn - CHEM - 225

University of Michigan-Dearborn - CHEM - 225

University of Michigan-Dearborn - CHEM - 225

University of Michigan-Dearborn - CHEM - 225

University of Michigan-Dearborn - CHEM - 225

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