Answer:
![y(t) = 2e^t -e^{-t}](https://tex.z-dn.net/?f=%20y%28t%29%20%3D%202e%5Et%20-e%5E%7B-t%7D%20)
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:
![ay'' +by' +cy=0](https://tex.z-dn.net/?f=%20ay%27%27%20%2Bby%27%20%2Bcy%3D0)
Where ![a =1, b=0, c=-1](https://tex.z-dn.net/?f=%20a%20%3D1%2C%20b%3D0%2C%20c%3D-1)
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:
![r^2 e^{rt} -e^{rt}=0](https://tex.z-dn.net/?f=%20r%5E2%20e%5E%7Brt%7D%20-e%5E%7Brt%7D%3D0)
We can take common factor
and we got:
![e^{rt} (r^2-1) =0](https://tex.z-dn.net/?f=%20e%5E%7Brt%7D%20%28r%5E2-1%29%20%3D0)
And for this case the two only possibel solutions are ![r=1, r=-1](https://tex.z-dn.net/?f=%20r%3D1%2C%20r%3D-1)
And the general solution for this case is given by:
![y = c_1 e^{r_1 t} + c_2 e^{r_2 t}](https://tex.z-dn.net/?f=%20y%20%3D%20c_1%20e%5E%7Br_1%20t%7D%20%2B%20c_2%20e%5E%7Br_2%20t%7D)
Replacing the roots that we found we got:
![y = c_1 e^{t} +c_2 e^{-t}](https://tex.z-dn.net/?f=%20y%20%3D%20c_1%20e%5E%7Bt%7D%20%2Bc_2%20e%5E%7B-t%7D)
Now we can find the derivates for this last espression
![y' = c_1 e^t -c_2 e^{-t}](https://tex.z-dn.net/?f=%20y%27%20%3D%20c_1%20e%5Et%20-c_2%20e%5E%7B-t%7D)
![y'' = c_1 e^t +c_2 e^{-t}](https://tex.z-dn.net/?f=%20y%27%27%20%3D%20c_1%20e%5Et%20%2Bc_2%20e%5E%7B-t%7D)
From the initial conditions we have this:
(1)
(2)
If we add equations (1) and (2) we got:
![4 = 2c_1 , c_1 = 2](https://tex.z-dn.net/?f=%204%20%3D%202c_1%20%2C%20c_1%20%3D%202%20)
And solving for
we got:
![c_2=3-c_1= 3-2 = 1](https://tex.z-dn.net/?f=%20c_2%3D3-c_1%3D%203-2%20%3D%201)
So then our general solution is given by:
![y(t) = 2e^t -e^{-t}](https://tex.z-dn.net/?f=%20y%28t%29%20%3D%202e%5Et%20-e%5E%7B-t%7D%20)