Consider the following initialvalue problem y 3y 0 y 0 1

Consider the following initial-value problem. y\" - 3y = 0, y (0) = 1, y\' (0) = 5 Solve the given problem first using the form of the general solution given in (10) in Section 4.3. y = c_1e^kx + C_2e^-kx y(x) = Solve again, this time using the form given in (11) in Section 4.3. y = c_1 cosh kx + C_2 sinh kx y(x) =

Solution

y\"-3y=0

auxiliary equation is given by D^2-3= 0

(D-3^1/2)(D+3^1/2)=0

D=3^1/2, -3^1/2

Complimentary function is given by

y(x) = Acosh(3^1/2)x+B Sinh(3^1/2)x

y(0)=1=A

y\'(0)=Asin0+Bcos0=B= 5

y(x)=Cosh(3^1/2)x+5 Sinh(3^1/2)x