Question

solve the initial value problem y''-4y'-5y=0 for y(0)=0 and y'(0)=6

Answer 1

Auxiliary equation,

m²-4m-5=0

solving the equation, we get

m=-1, 5

Therefore, general solution is

$y=c_1e^{-x}+c_2e^{5x}$

Then,

$y'=-c_1e^{-x}+5c_2e^{5x}$

By using initial conditions,

$y(0)=c_1+c_2=0\\ and\\ y'(0)=-c_1+5c_2=6$

solving above two equations, we get

$c_1=-1 ,c_2=1$

Therefore, solution of the given initial value problem is y=-e^-x +e^5x.