Answer:
For First Solution: 
is the solution of equation y''-y=0.
For 2nd Solution:
is the solution of equation y''-y=0.
Step-by-step explanation:
For First Solution: 
In order to prove whether it is a solution or not we have to put it into the equation and check. For this we have to take derivatives.

First order derivative:

2nd order Derivative:

Put Them in equation y''-y=0
e^t-e^t=0
0=0
Hence
is the solution of equation y''-y=0.
For 2nd Solution:

In order to prove whether it is a solution or not we have to put it into the equation and check. For this we have to take derivatives.

First order derivative:

2nd order Derivative:

Put Them in equation y''-y=0
cosht-cosht=0
0=0
Hence
is the solution of equation y''-y=0.