# One can describe any linear transformations with

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Problem 10) (10 points)Find the general solutions for the following differential equations:a) (5 points)f′′(t) +f(t) = 3 sin(2t) + 1, f(0) = 3, f(0) = 0b) (5 points)f′′(t)2f(t) +f(t) =t2, f(0) = 2, f(0) =4
Solution:
Problem 11) (10 points)Analyze the solutions (x(t), y(t)) for the following nonlinear dynamical systemddtx=xy2ddty=yxa) (3 points) Find the equations of the null-clines and find all the equilibrium points.b) (4 points) Analyze the stability of all the equilibrium points.c) (3 points) Which of the phase portraits A,B,C,D below belongs to the above system?
CDSolution:
Problem 12) (10 points)a) (4 points) Find the Fourier series of the functionf(x) =braceleftbiggπ,1x10,else.Minus1123456
b) (3 points) Use Parseval’s theorem to find the value of the sumsummationdisplayn=1sin2(n)n2.
c) (3 points) We add a parameter 0aπin the above computation:f(x) =braceleftbiggπ,axa0,else.What isg(a) =summationdisplayn=1sin2(na)n2?Solution:
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