MATH-M 344 Chapter 7: M344 7.6 Notes (Feb. 3)
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Can show linear independence with wronskian: general solution should be given in terms of real-valued functions. Taking complex conjugate of entire equation: ( (cid:2019)(cid:2869) )(cid:2778) =(cid:2777) (cid:2019)(cid:2869) =(cid:2019)(cid:2870) is also an e-value of , so (cid:2778) =(cid:2779) is a corresponding e-vector. Result is complex conjugate solutions (cid:2778)(cid:4666)(cid:4667)=(cid:2778)(cid:3090)(cid:3117) and (cid:2779)(cid:4666)(cid:4667)=(cid:2778) (cid:3090)(cid:3117) . Take real and imaginary parts of either (cid:2778) or (cid:2779) Find a fundamental set of real solutions of the system =[ (cid:2869)(cid:2870) (cid:2869)= (cid:2870) (cid:2869)=(cid:2870: to get set of real-valued solutions, by theorem 7. 4. 5, can choose real and imaginary parts of either. =(cid:4666) (cid:883),(cid:4667)=(cid:4666)(cid:883), (cid:4667: (cid:2778)(cid:4666)(cid:4667)=(cid:4672) (cid:3117)(cid:3118)+(cid:4673)(cid:4666)(cid:883),(cid:4667), (cid:2779)(cid:4666)(cid:4667)=(cid:4672) (cid:3117)(cid:3118) (cid:4673)(cid:4666)(cid:883), (cid:4667) (complex conjugates) (cid:2778) or (cid:2779, (cid:2778)(cid:4666)(cid:4667)= (cid:3118)(cid:4666)cos+sin(cid:4667)(cid:4666)(cid:883),(cid:4667) Imaginary part: (cid:2204)(cid:4666)(cid:4667)=(cid:4672) (cid:3118)sin, (cid:3118)cos(cid:4673)=(cid:4666) (cid:4667)(cid:3118)(cid:4666)sin,cos(cid:4667: to confirm linear independence, (cid:4666)(cid:2203),(cid:2204)(cid:4667)(cid:4666)(cid:4667)=| (cid:3118)cos. = (cid:882) for all: {(cid:2203),(cid:2204)} is fundamental set of solutions (cid:883) Give the general solution to the system =[(cid:882) (cid:883) (cid:882)] with initial condition (cid:4666)(cid:882)(cid:4667)=(cid:4666)(cid:884),(cid:883)(cid:4667): let =(cid:4666)(cid:2869),(cid:2870)(cid:4667, find e-values: solve | (cid:2019) (cid:883)
