Differential Calculus two parameter equation! Please help?

Question

Okay the question states

y=c1e^x + c2e^-1 is a two parameter family of solutions of the second-order DE y''-y=0. Find a solution of the second-order IVP consisting of this differential equation and the given initial conditions.

#1: y(-1)=5 y'(-1)=-5

I have no idea how to go about solving this If someone could guide me through this It would be greatly appreciated! Thanks

BTW the answer y=5e^(-x-1)

Answer 1

the solution to the ODE you have is:

y=c1 exp(x) +c2 exp(-x) where c1 and c2 are constants

you can evaluate those two constants when you are given two independent specific values of the function

the first initial condition tells you that when x=-1, the function equals 5; this means you substitute values to get:

5=c1exp(-1)+c2exp(1)  eq. (1)

the second condition means that the first derivative has the value of -5 when x=-1; so take the first derivative:

y'=c1exp(x)-c2exp(-x)  and substitute values:

-5=c1exp(-1)-c2exp(1) eq. (2)

you now solve eqs. (1) and (2) to find the values of c1 and c2, and subsitute these values into the solution above

hope this helps get you started