Math 2280 - Final ExamUniversity of UtahSpring 2009Name:1Laplace Transforms You May NeedDefinitionL(f(t)) =Z∞0e−stf(t)dt.L(eat) =1s − aL(sin (kt)) =ks2+ k2L(cos (kt)) =ss2+ k2L(δ(t − a)) = e−asL(u(t − a)f(t − a)) = e−asF (s).Eigenvalue Rules for Critical Pointsλ1< λ2< 0 Stable improper nodeλ1= λ2< 0 Stable node or spiral pointλ1< 0 < λ2Unstable saddle pointλ1= λ2> 0 Unstable node or spiral pointλ1> λ2> 0 Unstable improper nodeλ2, λ2= a ± bi, (a < 0) Stable spiral pointλ1, λ2= a ± bi, (a > 0) U nstable spiral pointλ1, λ2= ±bi Stable or unstable, center or spiral point2Fourier Series DefinitionFor a function f (t) of period 2L the Fourier series is:a02+∞Xn=1ancosnπtL+ bnsinnπtL.an=1LZL−Lf(t) cosnπtLdtbn=1LZL−Lf(t) sinnπtLdt.3Basic Definitions (5 points)Circle or state the correct a nswer for the questions about the follow-ing differential equa tion:x2y′′− sin (x)y′+ y3= e2x(1 point) The differential equation is: Linear Nonlinear(1 points) The order of the differential equation is:For the differential equation:(x4− x)y(3)+ 2xexy′− 3y =px − cos (x)(1 point) The differential equation is: Linear Nonlinear(1 point) The order of the differential equation is:(1 point) The corresponding homogeneous equation is:4Separable Equations (5 points)Find the general solution to the differential equation:dydx= 3√xy5Linear First-Order Equations (5 points)Find the particular solution to the differential equation below withthe given value:xy′+ 3y = 2x5;y(2) = 1.6Continued...7Higher Order Linear Differential Equations (5 points)Find the general solution to the linear differential equation:y′′− 3y′+ 2y = 0.8Nonhomogeneous Linear Differential Equations (10 points)Find the general solution to the differential equation:y(3)+ 4y′= 3x − 1.9Continued...10Systems of Differential Equations (10 points)Find the general solution to the system of differential equations:x′1= 5x1+ x2+ 3x3x′2= x1+ 7x2+ x3x′3= 3x1+ x2+ 5x3Hint : λ = 2 is an eigenvalue of the coefficient matrix, and all e igen-values are real.11Continued...12Continued...13Systems of Differential Equations with Repeated Eigenvalues (5 points)Find the general solution to the system of differential equations:x′=1 −44 9x.14Continued...15Laplace Transforms (5 points)Using the definition of the Laplace transform calculate the Laplacetransform of the function:f(t) = e3t+1.16Laplace Transforms and Differential Equations (8 points)Find the particular solution to the differential equation:x′′+ 4x = δ(t) + δ(t − π);x(0) = x′(0) = 0.17Continued...18Nonlinear Systems (7 points)Determine the location of the critical point (x0, y0) for the systemgiven below, and classify the critical point as to its type and stability.dxdt= x + y − 7,dydt= 3x − y − 5.19Continued...20More Nonlinear Systems (10 points)For the nonlinear system below, determine all critical points, andclassify each according to its type and stability.dxdt= 3x − x2+12xy,dydt=15xy − y.21Continued...22Continued...23Ordinary, Regular, and Irregular Points (5 points)Determine if the point x = 0 in the following second order differ-ential equation is an ordinary point, a regular singular point, or anirregular singular point.x3y′′+ 6 sin (x)y′+ 6xy = 0.24Power Series S olutions (10 points)Find a general solution in powers of x to the differential equation:(x2+ 1)y′′+ 6xy′+ 4y = 0.25Continued...26Continued...27Fourier Series (10 points)The values of the periodic function f(t) in one full period are given.Find the function’s Fourier series.f(t) =−1 −2 < t < 01 0 < t < 20 t = {−2, 0}Extra Credit (2 points) - Use this solution and what you know aboutFourier series to de d uce the famous Leibniz formula for
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