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Consider the differential equation associated with y = Σ (from i=1 to 3) C_i e^(m_i x), where C_i represents arbitrary constants and m₁, m₂, m₃ are the solutions of m³ - 7m + 6 = 0. If the equation is expressed as d³y/dx³ - 7 dy/dx + k y = 0, determine the value of k.
- 6
- 7
- 8
- 9
Correct answer: 6
Solution
Since m1,m2,m3 are the roots of m^3 - 7m + 6 = 0, the linear ODE with general solution y = sum C_i e^(m_i x) is (D^3 - 7D + 6)y = 0. Comparing with d^3y/dx^3 - 7 dy/dx + k y = 0 gives k = 6, option (a).
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