# 4.3 Notes - Slide 1 Homogeneous Linear Equations with...

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Slide 1 Homogeneous Linear Equations with Constant Coefficients Differential Equations Section 4.3 Slide 2 Homogeneous Linear Equations with Constant Coefficients A differential equation of the form can always be solved in terms of elementary functions of calculus. If the coefficients are not constant they are usually much more difficult. 0 = + + cy y b y a Slide 3 Example Can you think of a solution of the differential equation given below? How many different solutions can you come up with just using trial and error? 0 = - y y Possibilities for y: e t , e -t , ce t , ce -t , A linear combination thereof

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Slide 4 General Case What about the equation ? We will form the auxiliary equation am 2 + bm + c = 0 . There are three possibilities for the roots m 1 and m 2 of this equation: m 1 and m 2 are real and distinct m 1 and m 2 are real and equal m 1 and m 2 are conjugate complex numbers 0 = + + cy y b y a Slide 5 Case I: Distinct Real Roots In this case we find two solutions,
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4.3 Notes - Slide 1 Homogeneous Linear Equations with...

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