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# 4.3 Notes - Slide 1 Error Objects cannot be created from...

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Unformatted text preview: Slide 1 Error! Objects cannot be created from editing ﬁeld codes. Homogeneous Linear Equations with Constant Coefficients Section 4.3 Slide 2 Error! Objects cannot be created from editing ﬁeld codes. 4 Homogeneous Linear Equations with Constant Coefficients A differential equation of the form ay'+by’+cy =0 can always be solved in terms of elementary functions of catculus. If the coefﬁcients are not constant they are usually much more difﬁcult. Slide 3 Error! Objects cannot be created from editing ﬁeld Possibilities for y: e‘, e", ce‘. ce", A linear codes. combination thereof Example Can you think of a solution of the differential equation given below? y'—y=0 How many different solutions can you come up with just using trial and error? Slide 4 Error! Objects cannot be created from editing ﬁeld codes. General Case What about the equation ay'+by'+cy=0 ‘? 5‘ Wewill ton-n the auxiliary equation am2 + om +c = 0. - There are three possibilities for the roots m, and mi of this " eqoation: o m, and mzare {ml and distinct o m, and mzare real and equal I m, and m, are conjugate complex numbers Slide 5 Error! Objects cannot be created from editing ﬁeld codes. ’ Case I: Distinct Real Roots In this case we find two solutions, yl = 9"” and y: = 3"“ ~ These functions are linearly independent on (-~,-=) and hence form a fundamental set of solutions. 2' f Any equation ofthe feiTn y = 019’"1 +czem “here m| and n12 are distinct real solutions of the auxiliary equation I will be a solution of the given differential equation. Exanple: 4y”; 9y = 0 Slide 6 Error! Objects cannot be created from editing ﬁeld codes. I Case ll: Repeated Real Roots in thistmewe find hmsolutions. yl :e'“ and y2 = xe’""‘ These functions are linearly independent on (--°,-“) and hence form a fundamental set of solutions. Any equation diheform y = clem'r +c2xem“ where m1: m2 are repeated real solutions of the auxiliary equation will be a solution of the given differential equation. Exam”? y'—10y'+25y =0 \ Slide 7 Error! Objects cannot be created from editing ﬁeld codes. Case lll: Conjugate Complex Roots In this case we ﬁnd Me solutlons. __, e-urand = ‘1‘ wh nd y] y: e ereml=a+ﬂia Minx—ﬁr" Using Euler‘s formula we can rewrite these solmions as y.=2e‘”cos,8x and y,=22‘"sin,8x- These functions as linearly inde dent on (my) and hence form a ‘ fundamental set of solutions, so he general solution will be y = ole“ oosﬁx+czrfn sin ﬁx ExamP'e: y'w2y'+2y=0, y'(0)=l, y(0)=1 Slide 8 Error! Objects cannot be created from editing ﬁeld codes. Higher-Order Equations 0 Form the auxiliary equation and find all the roots. You may have to use synthelic division. 0 The methods of writing the solutions of the diﬁerenlial equaiions on the previous slide are just extended. Refer to pages 145 and 146 in your text. " 9 Example: y'+ 2)" -5y' - 6y = 0, y(0) = y'(0) = 0, y'(0) = 1 ...
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