Dierential Equations: Final ReviewJuly 25, 20121stOrder Linear EquationSolve the dierential equation1)y′+ y2sin (x) = 02)t3y′+ 4t2y = e−t3)(2x + 3) + (2y − 2)y′= 04)(3x2y + 2xy + y3)dx + (x2+ y2)dy = 0Find the amount of salt in the tank at timet.5) A tank contains100gal of water and50oz of salt. Water containing a salt concentration of14(1 +12sin (t))oz/gal ows into the tank at a rate of2gal/min, and the mixture in the tank ows out at the samerate.Systems of Two First Order Equations1) Create a chart with the following columns: Conditions onλ(the eigenvalues of the matrix), type (name)of system, stability, sketch of typical phase portrait.Solve the system forxand draw the phase portrait.2) Real, distinctλ:x′=(1 14 −2)x3) Real, distinctλ:x′=(−2 11 −2)x4) Complex pair:x′=(−1 −41 −1)x5) Repeatedλ:x′=(−1 −122 −3)x6) Non-homogeneous:x′=(−1 −20 1)x +(e−tt)1Second Order Linear EquationsSolve the following equations:1)2y′′+ 3y′+ y = t2+ 3sin (t)2)y′′+ y = tan (t)Laplace TransformsFind the Laplace Transform of:1)f(t) = e3tcos (2t)2)f(t) ={0 t < 1t2− 2t + 2 t ≥ 13)f(t) = tfor0 ≤ t < 1; andf(t)has period14)f(t) =´t0(t − τ)2cos (2τ) dτFind the inverse Laplace of:5)F (s) =2s+2s2+2s+56)F (s) =2(s−1)e−2ss2−2s+2Solve the dierential equation7)y′′+ 2y′+ 5y = e−tsin (2t)wherey(0) = 2andy′(0) = −18)y′′+ y = δ(t − 2π) cos (t)wherey(0) = 0andy′(0) = 1Systems of First Order Linear EquationsFind a fundamental matrix for the given system:1) (defective matrix):x′=−7 9 −6−8 11 −7−2 3
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