Answer:

Step-by-step explanation:
Assuming this complete problem: "In this problem,
y = c1ex + c2e−x
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.
y(0) = 1, y'(0)= 3"
Solution to the problem
For this case we have a homogenous, linear differential equation with order 2, and with the general form:

Where 
And we can rewrite the differential equation in terms
like this:
![[e^{rt}]'' -e^{rt}=0](https://tex.z-dn.net/?f=%20%5Be%5E%7Brt%7D%5D%27%27%20-e%5E%7Brt%7D%3D0)
And applying the second derivate we got:

We can take common factor
and we got:

And for this case the two only possibel solutions are 
And the general solution for this case is given by:

Replacing the roots that we found we got:

Now we can find the derivates for this last espression


From the initial conditions we have this:
(1)
(2)
If we add equations (1) and (2) we got:

And solving for
we got:

So then our general solution is given by:
