Math 2280 - Exam 2University of UtahSpring 2009Name:1Existance and Uniqueness - State whether we’re certain (based on our ex-istance and uniqueness theorem for linear differential equations) aunique solution exists for the following differential equations on thegiven interval. Explain why. (5 points)1. (1 point)y′′− x(y′)2+ exy = 2x2− 5;for all x ∈ R.2. (2 points)xy′′− exy′+ cos xy = 25x3;for all x > 1.23. (2 points)xy′′− exy′+ cos xy = 25x3;for all x < 1.3Linear Differ e ntial Equations with Constant Coefficients (10 points)1. Find the general solution to the following homogeneous differ-ential equation: (3 points)y′′− y′− 6y = 042. Use this result to calculate the general solution to the nonhomo-geneous differential equation: (4 points)y′′− y′− 6y = 2x + e−2x53. Find the unique solution to the following initial value problem:(3 points)y′′− y′− 6y = 2x + e−2xy(0) = 2 , y′(0) =7156Wronskians - Calculate the Wronskian for the following sets of functions,and determine if the functions are linearly indep endent. If the func-tions are not linearly independent, demonstrate a non-trivial linearcombination that equals 0. (5 points)1. (2 points)y1= e3xy2= xe3x72. (3 points)y1= sin 2xy2= sin x cos x8Converting to First-Order Systems - Convert the following system of equa-tions into an equivlent system of first-order equations: (5 points)x(3)= x′′− 2x′+ 5y′+ 2x + 1y′′= x′+ 5x − 14y′9Circuits Calculate the steady periodic current for the circuit pictured be-low: (10 points)with the following parameters: R = 200Ω, L = 5H, C = .001F , andE(t) = 100 sin (10t)V .10Continued...11First-Order Systems Solve the system of first-order differential equationsgiven below: (10 points)x′1= 3x1+ x2+ x3x′2= −5x1− 3x2− x3x′3= 5x1+ 5x2+ 3x312Method of U n d etermined Coefficients Find the form of the particular so-lution (but don’t calculate the constants) for the nonhomogeneouslinear differential equation given below using the me thod of unde-termined coefficients: (5 points)y(4)− 2y′′+ 3y′− 10y = x3e−xcos
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