Question

how to find auxillary equation in differential equations ​

Answer 1

$To \: clarify \: the \: question\: about \: linearity :$

$Both \: equations$

$y^{..} - 4y^{.}+16y=0$

$2x^{2}y^{..}+5xy^{.}+y=0$

$are\: linear.$

$for \:ODEs, \: linear \: means \: linear \: relatively\: to \: y, \\ \: y^{.},\: y^{..} \: even \: if \: the \: coefficients \: of \: them \: are \\ \: not \: linear.$

$to \: solve - \: {y}^{..} - 4 {y}^{.} + 16y = 0$

$change \: of \: function - \: y(x) = ce^{mx}$

$in \: order \: to \: obtain \: - \: {m}^{2} - 4m + 16$

$to \: solve \: - \: 2 {x}^{2} y ^{..} + 5x {y}^{.} = 0$

$change \: in \: function - \: y(x) = c x^{m}$

$in \: order \: to \: obtain - \: 2m(m - 1) + 5m + 1 = 0$