Spectral decomposition � 1 u 1 u 1 T � 2 u 2 u 2 T � n u n u n T Second order

Spectral decomposition ? 1 u 1 u 1 t ? 2 u 2 u 2 t ?

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Spectral decomposition:λ1u1u1T+λ2u2u2T+· · ·+λnununTSecond-order and Higher-order differentialequationsHomogeneous solutions:Auxiliary equation: Replace equation bypolynomial, soy000becomesr3etc. Then find the zeros (use the rationalroots theorem and long division, see the ‘Diagonalization-section).’Simple zeros’ give youert, Repeated zeros (multiplicitym) give youAert+Btert+· · ·Ztm-1ert, Complex zerosr=a+bigive youAeatcos(bt) +Beatsin(bt).Undetermined coefficients:y(t) =y0(t) +yp(t), wherey0solves thehom. eqn. (equation= 0), andypis a particular solution. To findyp:If the inhom. term isCtmert, then:yp=ts(Amtm· · ·+A1t+ 1)ert, where ifris a root of aux withmultiplicitym, thens=m, and ifris not a root, thens= 0.If the inhom term isCtmeatsin(βt), then:yp=ts(Amtm· · ·+A1t+ 1)eatcos(βt) +ts(Bmtm· · ·+B1t+ 1)ertsin(βt), wheres=m, ifa+biis also a root of aux with multiplicitym(s= 0if not).cosalways goes withsinand vice-versa, also, you have to look ata+bias one entity.Variation of parameters:First, make sure the leading coefficient(usually the coeff. ofy00) is= 1.. Theny=y0+ypas above. Nowsupposeyp(t) =v1(t)y1(t) +v2(t)y2(t), wherey1andy2are yourhom. solutions. Theny1y2y01y02v01v02=0f(t). Invert the matrix andsolve forv01andv02, and integrate to getv1andv2, and finally use:yp(t) =v1(t)y1(t) +v2(t)y2(t).Useful formulas:abcd-1=1ad-bcd-b-caRsec(t) = ln|sec(t) + tan(t)|,Rtan(t) = ln|sec(t)|,Rtan2(t) = tan(x)-x,Rln(t) =tln(t)-tLinear independence:f, g, hare linearly independent ifaf(t) +bg(t) +ch(t) = 0)a=b=c= 0. To show linear
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dependence, do it directly. To show linear independence, form theWronskian:fW(t) =f(t)g(t)f0(t)g0(t)(for2functions),fW(t) =24f(t)g(t)h(t)f0(t)g0(t)h0(t)f00(t)g00(t)h00(t)35(for3functions). Then pick apointt0wheredet(fW(t0))is easy to evaluate. Ifdet6= 0, thenf, g, hare linearly independent! Try to look for simplifications before youdifferentiate.Fundamental solution set:Iff, g, hare solutionsandlinearlyindependent.Largest interval of existence:First make sure the leading coefficientequals to1. Then look at the domain of each term. For each domain,consider the part of the interval which contains the initial condition.Finally, intersect the intervals and change any brackets to parentheses.Harmonic oscillator:my00+by0+ky= 0(m=inertia,b=damping,k=stiffness)Systems of differential equationsTo solvex0=Ax:x(t) =Aeλ1tv1+Beλ2tv2+eλ3tv3(λiareyour eigenvalues,viare your eigenvectors)Fundamental matrix:Matrix whose columns are the solutions, withoutthe constants (the columns are solutions and linearly independent)Complex eigenvaluesIfλ=+iβ, andv=a+ib. Then:x(t) =A(etcos(βt)a-etsin(βt)b)+B(etsin(βt)a+etcos(βt)b)Notes:You only need to consider one complex eigenvalue. For realeigenvalues, use the formula above. Also,1a+bi=a-bia2+b2Generalized eigenvectors
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