y5y6y0 y09 y03 y04 yxSolutiony5y6y 0 let yv v 5v6v 0 cha

y5y+6y=0,

y(0)=9,  y(0)=3,  y(0)=4.

y(x)=

Solution

y\'\'\'-5y\"+6y\' = 0

let y\'=v

=>

v\" -5v\'+6v = 0

=>

charecteristic equation is

m^2-5m+6 =0

=>

m = 2,3

=>

v = c1e^(2x) +c2e^(3x)

=>

y\' = c1e^2x +c2e^3x

y\'(0) = 3 => c1+c2 = 3

y\"(0) = 4 => 2c1+3c2 = 4

=>

c2 = -2, c1 = 5

=>

y\' = 5e^2x -2e^3x

=>

y = 5e^2x /2 - 2e^3x /3 +c

y(0) = 9

=>

5/2 -2/3 +c = 9

=>

c = 9-11/6 = 43/6

=>

y = 5e^2x /2 - 2e^3x /3 +43/6

=>

y = (15e^2x -4e^3x +43)/6