MATH-S 343 Chapter Notes - Chapter 3: Equating Coefficients, Linear Combination, General Idea
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S343 section 3. 5 notes- nonhomogeneous equations; method of undetermined coefficients. 10-13-16 nonhomogeneous ode: proof: homogeneous ode, proof: Because all solutions can be expressed as linear combinations of fundamental sets of solutions. Theorem 3. 5. 2- the general solution of the nonhomogeneous ode can be written as. Nonhomogeneous equation- []= +(cid:1868)(cid:4666)(cid:1872)(cid:4667) +(cid:1869)(cid:4666)(cid:1872)(cid:4667)=(cid:4666)(cid:1872)(cid:4667) where (cid:1868),(cid:1869), are given continuous functions on open interval: (cid:4666)(cid:1872)(cid:4667)=(cid:882) gives homogeneous equation +(cid:1868)(cid:4666)(cid:1872)(cid:4667) +(cid:1869)(cid:4666)(cid:1872)(cid:4667)=(cid:882) Theorem 3. 5. 1- suppose (cid:2869)(cid:4666)(cid:1872)(cid:4667) and (cid:2870)(cid:4666)(cid:1872)(cid:4667) are solutions of the nonhomogeneous ode such that. +(cid:1868)(cid:4666)(cid:1872)(cid:4667) +(cid:1869)(cid:4666)(cid:1872)(cid:4667)=(cid:4666)(cid:1872)(cid:4667)=(cid:2869) or (cid:2870); then (cid:2870) (cid:2869)=(cid:4666)(cid:1872)(cid:4667) (cid:4666)(cid:1872)(cid:4667)=(cid:882) is a solution to the. If {(cid:2869),(cid:2870)} is also a fundamental set of solutions for the homogeneous equation, then (cid:1855)(cid:2869)(cid:2869)(cid:4666)(cid:1872)(cid:4667)+(cid:1855)(cid:2870)(cid:2870)(cid:4666)(cid:1872)(cid:4667)=(cid:2869)(cid:4666)(cid:1872)(cid:4667) (cid:2870)(cid:4666)(cid:1872)(cid:4667)=(cid:882) So [(cid:2869) (cid:2870)](cid:4666)(cid:1872)(cid:4667)=(cid:882), which shows (cid:2869) (cid:2870) is solution to homogeneous ode (theorem 3. 2. 4), we can write (cid:2869) (cid:2870)=(cid:1855)(cid:2869)(cid:2869)+(cid:1855)(cid:2870)(cid:2870)=(cid:882) =(cid:4666)(cid:1872)(cid:4667)=(cid:1855)(cid:2869)(cid:2869)(cid:4666)(cid:1872)(cid:4667)+(cid:1855)(cid:2870)(cid:2870)(cid:4666)(cid:1872)(cid:4667)+(cid:4666)(cid:1872)(cid:4667), where (cid:2869),(cid:2870) are a fundamental set of solutions to corresponding homogeneous ode, (cid:1855)(cid:2869),(cid:1855)(cid:2870) are arbitrary constants, and is some specific solution of the.
