2.7 Homogeneous Diff Eq. 1

2.7 Homogeneous Diff Eq. 1 - 2.7 2.7 Homogeneous Linear...

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2.7. 2.7. Homogeneous Linear Differential Equations with Con- stant Coefficients 1. (a) T (p.137 corollary to Theorem 2.32) (b) T (p.132 Theorem 2.28) (c) F (d) F (Any solution is a linear combination of e at and t k e at ) (e) T ( ) If x and y are solutions of p ( D ) = 0, then p ( D )( αx + βy ) = αp ( D ) x + βp ( D ) y = 0 + 0 = 0 ,α,β F αx + is a solution of p ( D ) = 0 (f) F ( ) It’s different with the multiplicity of c i (p.137 and 139, Theorem 2.33 and 2.34) (g) T (p.131) 2. (a) F Let S = { a 1+ t 2 | a R } ⇒ S : 1-dimensional subspace of C But there is no homogeneous linear differential equation with constant coefficients (b) F Let { t,t 2 } is the solution of y 00 + ay 0 + by = 0 0 + a + bt = 0 a = b = 0 140 PNU-MATH
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2.7. then y 00 + ay 0 + by = 0 becomes y 00 = 0 ( t 2 ) 00 = 2 = 0 (cf) y 000 = 0 D 3 = 0 t = 0 e 0 t ,te 0 t ,t 2 e 0 t i.e. 1 ,t,t 2 y 000 = 0 (c) T Let x is a solution to the homogeneous linear differential equation with constant coefficients P ( D ) y = 0 Since P ( D ) x = 0 , P ( D ) x 0 = P ( D )( Dx ) = P ( D ) Dx = DP ( D ) x = D (0) = 0 x 0 is also a solution to the equation (d)T Let p ( D ) x = 0 and q ( D ) y = 0 p ( D ) q ( D )( x + y ) = p ( D ) q ( D ) x + p ( D ) q ( D ) y = q ( D
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2.7 Homogeneous Diff Eq. 1 - 2.7 2.7 Homogeneous Linear...

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