Question

(D^4 - D^3 - 9D^2 - 11D - 4)y = 0​

Answer 1

ANSWER

$given \: differential \: equation \: is \\( D {}^{4} \: - D {}^{3} \: - 9D {}^{2} \: - 11D \: - 4 \: )y = 0 \\ \\ its \: auxiliary \: equation \: is \\ \\ m {}^{4} - m {}^{3} - 9m {}^{2} - 11m - 4 = 0 \\ \\ (m + 1)(m {}^{3} -2 m {}^{2} - 7m - 4) = 0 \\ \\( m + 1) \: \: (m {}^{3} - 2m {}^{2} - 7m - 4) = 0 \\ \\ (m + 1)(m + 1)(m {}^{2} - 3m - 4) = 0 \\ \\ (m + 1)(m + 1 )(m - 4)(m + 1) = 0 \\ \\ therefore \: \: m = - 1 \: - 1 \: - 1 \: \: 4 \\ \\ hence \: \: c.f \: \: of \: given \: differential \: equation \: is \\ \\ c.f \: \: = (k1 + k2x + k3x {}^{2} )e {}^{ - 1} + k4e {}^{4} \\ \\ where \: \: k1 \: \: k2 \: \: k3 \: \: k4 \: \: are \: real \: constant$