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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 ay by cy 0 can always be solved in terms of elementary functions of calculus. If the coefficients are not constant they are usually much more difficult. Possibilities for y: et, e-t, cet, ce-t, A linear combination thereof Slide 3 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 General Case What about the equation ay by cy 0 ? We will form the auxiliary equation am2 + bm + c = 0. There are three possibilities for the roots m1 and m2 of this equation: m1 and m2 are real and distinct m1 and m2 are real and equal m1 and m2 are conjugate complex numbers Slide 5 Case I: Distinct Real Roots In this case we find two solutions, y1 e m1x and y2 e m2 x These functions are linearly independent on (-∞,∞) and hence form a fundamental set of solutions. y c1e c2e Any equation of the form where m1 and m2 are distinct real solutions of the auxiliary equation will be a solution of the given differential equation. m1t Example: m2t 4 y 9 y 0 Slide 6 Case II: Repeated Real Roots In this case we find two solutions, y1 e m1x and y2 xem1x These functions are linearly independent on (-∞,∞) and hence form a fundamental set of solutions. y c1e 1 c2 xe 1 Any equation of the form where m1 = m2 are repeated real solutions of the auxiliary equation will be a solution of the given differential equation. mt Example: y 10 y 25 y 0 mt Slide 7 Case III: Conjugate Complex Roots In this case we find two solutions, y e m1x and y e m2 x 1 2 where m i and m i . 2 1 Using Euler's formula we can rewrite these solutions as y 2e x cos x and y 2e x sin x . 1 2 These functions are linearly independent on (-∞,∞) and hence form a fundamental set of solutions, so the general solution will be y c1e x cos x c2e x sin x Example: y 2 y 2 y 0, y(0) 1, y (0) 1 Slide 8 Higher-Order Equations Form the auxiliary equation and find all the roots. You may have to use synthetic division. The methods of writing the solutions of the differential equations on the previous slide are just extended. Refer to pages 145 and 146 in your text. Example: y 2 y 5 y 6 y 0, y (0) y(0) 0, y(0) 1