Find a member of the family that is a solution of the initial-value problem:

Question

anonymous · Answer

$$y = c_{1}e ^{x} + c _{2}e ^{-x}, (-\infty, \infty);$$
$$y \prime \prime - y = 0,   y(0) = 0,   y \prime(0) = 1$$

turingtest · Answer

they just want you to find c1 and c2
have you never done this before?

turingtest · Answer

oh wait, a double derivative and only single condition...

anonymous · Answer

Well, I kinda skipped this section in class. I can find general solutions, but the wording here throws me

turingtest · Answer

still though, it should work...

anonymous · Answer

so what just plug in values and take integrals?

turingtest · Answer

plug in the conditios given
y(0)=?
you will get an equation
y'(0)=?
you will get another equation

two equations with two unknowns; solve the system for the unknowns

anonymous · Answer

so is all that find the member family talk just find the values of c1 and c2?

turingtest · Answer

the value of c1 and c2 you get (which depends on the initial conditions) determines which "member of the family", so to speak, is the answer you are searching for
by leaving c1 and c2 as unknown, it is general (the whole family)
by using initial condition to find c1 and c2 you have found a specific solution ( a member of the family)

turingtest · Answer

so the problem is asking you to use the initial conditions to find c1 and c2, which will tell you which member of the family of integral curves is the solution to our IVP

anonymous · Answer

So is this similar to finding the particular solution of a differential equation, or something totally different?

turingtest · Answer

ok let me double-check my terminology...

turingtest · Answer

yes, as I suspected the terminology is a bit tricky here
the "particular solution" from a non-homogeneous equation found through things like undetermined coefficients and such) is not to be confused with the solution to out initial value problem, in which we identify the values of c1 and c2 to determine which integral curve we are talking about
I know of no other term for "the integral curve" or "member of the solution family" for what I am referring to, but the "particular solution to a non-homogeneous equation" is a very different thing.

anonymous · Answer

awesome, you are a champ. Im going to try out just solving. Thanks a bunch for the clarity.

turingtest · Answer

very welcome :)