To determine An ODE for the given basis in the form of y ″ + a y ′ + b y = 0 . Explanation Result used: Case (i): If the auxiliary equation has distinct real roots λ 1 , and λ 2 of y = e λ x , then the basis of y ″ + a y ′ + b y = 0 is e λ 1 x , e λ 2 x and the general solution of y ″ + a y ′ + b y = 0 is y = c 1 e λ 1 x + c 2 e λ 2 x . Case (ii): If the auxiliary equation has real double root λ = − a of y = e λ x , then the basis of y ″ + a y ′ + b y = 0 is e − a x , x e − a x and the general solution of y ″ + a y ′ + b y = 0 is y = ( c 1 + c 2 ) e − a x . Case (iii): If the auxiliary equation has complex conjugate root λ 1 = a + i ω , λ 2 = a − i ω of y = e λ x , then the basis of y ″ + a y ′ + b y = 0 is e a x cos ω x , e a x sin ω x and the general solution of y ″ + a y ′ + b y = 0 is y = e − a x ( A cos ω x + B sin ω x ) . Calculation: The given basis is e − 3.1 x cos 2.1 x , e − 3.1 x sin 2.1 x , that is, the ODE has complex conjugate root. It is known that, the general form of the complex number is a + i b . Here, by case (iii), a = ‑3.1 and b = 2.1. Obtain an ODE y ″ + a y ′ + b y = 0 for the given basis as follows

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Author: Kreyszig

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Chapter 2.2, Problem 20P

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An ODE for the given basis in the form of y″+ay′+by=0.

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Chapter 2.2, Problem 19P

Chapter 2.2, Problem 21P

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