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Find the general solution. D3(D − 2)(D − 3)2y = 0. |
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Answer» The characteristic polynomial is λ3(λ − 2)(λ − 3)2. Thus, λ = 0 is a root of multiplicity 3, so it contributes the basic solutions e0x, xe0x, x2e0x, i.e., 1, x, x2. We have λ = 2 as a root of multiplicity 1, so it contributes the basic solution e2x. Finally, we have λ = 3 as a root of multiplicity 2, so it contributes the basic solutions e3x, xe3x. The general solution of the equation is a linear combination, with arbitrary coefficients, of the basic solutions, so the general solution is y = C1 + C2x + C3x2 + C4e2x + C5e3x + C6xe3x. |
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