Find y as a function of x if y3yy3y0 y01 y07 y031 yxSolution

Find y as a function of x if

y3yy+3y=0,

y(0)=1,  y(0)=7,  y(0)=31.
y(x)=

Solution

charecteristic equaiton is

m^3 -3m^2 -m +3 = 0

=>

(m-1)(m+1)(m--3) = 0

=>

m = -1,1,3

=>

general solution is

y = c1e^(-x) +c2e^x +c3e^3x

y(0) = 1

=>

c1+c2 +c3 = 1..........(1)

y\'(0)= 1

=>

-c1 +c2 +3c3 = 7 .......(2)

y\"(0) = -31

=>

c1+c2 +9c3 = -31....(3)

(1,),(2),(3) implies

c1 =-9,c2 =16, c3=-6

=>

y = -9e^(-x) +16e^x -6e^3x