homework set 16
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homework set 16

Course Number: COMPUTER S Math 225, Spring 2010

College/University: Bilkent University

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BILKENT UNIVERSITY 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...

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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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Bilkent University - COMPUTER S - Math 225
BILKENT UNIVERSITY Department of Mathematics MATH 225, LINEAR ALGEBRA and DIFFERENTIAL EQUATIONS, Solution of Homework set1 # 16 U. Muan g July 16, 20081) a) W (e2x , e-3x/2 ) = b) W (x, xex ) = c) W (ex sin x, ex cos x) = ex sin x ex cos x = -e2x ex si
Bilkent University - COMPUTER S - Math 225
BILKENT UNIVERSITY Department of Mathematics MATH 225, LINEAR ALGEBRA and DIFFERENTIAL EQUATIONS, Homework set # 17 U.Muan g July 16, 2008REPEATED ROOTS; REDUCTION OF ORDER 1) a) b) c) 2) a) b) c) 3) a) b) c) 4) Find the general solution of the followin
Bilkent University - COMPUTER S - Math 225
BILKENT UNIVERSITY Department of Mathematics MATH 225, LINEAR ALGEBRA and DIFFERENTIAL EQUATIONS, Solution of Homework set1 # 17 U. Muan g July 16, 20081) a) Substituting y = erx into the D.E., we get the C.E. r2 - 2r + 1 = 0 which gives r1 = r2 = 1. Th
Bilkent University - COMPUTER S - Math 225
BILKENT UNIVERSITY Department of Mathematics MATH 225, LINEAR ALGEBRA and DIFFERENTIAL EQUATIONS, Homework set # 18 U.Muan g July 16, 2008METHOD OF UNDETERMINED COEFFICIENTS 1) Find the general solution of the following D.E's a) y - 2y - 3y = 3e2x . b)
Bilkent University - COMPUTER S - Math 225
BILKENT UNIVERSITY Department of Mathematics MATH 225, LINEAR ALGEBRA and DIFFERENTIAL EQUATIONS, Solution of Homework set1 # 18 U. Muan g 1) a) First we find the solution of the corresponding homogenous D.E. y - 2y - 3y = 0, which has the C.E. r2 - 2r -
Bilkent University - COMPUTER S - Math 225
BILKENT UNIVERSITY Department of Mathematics MATH 225, LINEAR ALGEBRA and DIFFERENTIAL EQUATIONS, Homework set # 19 U.Muan g July 16, 2008VARIATION OF PARAMETERS 1) Find a particular solution of the following D.E. by using the variation of parameters. T
Bilkent University - COMPUTER S - Math 225
BILKENT UNIVERSITY Department of Mathematics MATH 225, LINEAR ALGEBRA and DIFFERENTIAL EQUATIONS, Solution of Homework set1 # 19 U. Muan g 1) a) Corresponding homogenous equation is y - y - 2y = 0 Since, it is a constant coefficient D.E. we look for the
Bilkent University - COMPUTER S - Math 225
BILKENT UNIVERSITY Department of Mathematics MATH 225, LINEAR ALGEBRA and DIFFERENTIAL EQUATIONS, Solution of Homework set1 # 20 U. Muan g July 4, 2008Homework problems from the 2nd Edition, SECTION 6.13(3)2 The characteristic eq. p() = 2 - 7 + 10 = 0.
Bilkent University - COMPUTER S - Math 225
BILKENT UNIVERSITY Department of Mathematics MATH 225, LINEAR ALGEBRA and DIFFERENTIAL EQUATIONS, Solution of Homework set1 # 21 U. Muan g July 4, 2008Homework problems from the 2nd Edition, SECTION 6.215(15)2 The characteristic eq. p() = -3 + 22 - = -
Bilkent University - COMPUTER S - Math 225
BILKENT UNIVERSITY Department of Mathematics MATH 225, LINEAR ALGEBRA and DIFFERENTIAL EQUATIONS, Solution of Homework set1 # 22 U. Muan g July 4, 2008Homework problems from the 2nd Edition, SECTION 6.315(15)2 The C.E. p() = 2 - 3 + 2 = ( - 2)( - 1)2 =
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