Find out some solutions to

y′′ - 9 y = 0

Solution

 We can find some solutions here simply through inspection. We require functions whose second derivative is 9 times the original function. One of the first functions which I can think of that comes back to it after two derivatives is an exponential function and along with proper exponents the 9 will find taken care of as well.

Therefore, it looks like the subsequent two functions are solutions.

y(t) = e^3t and    y(t) = e^-3t

We'll leave this to you to verify that these are actually solutions.

These two functions are not the merely solutions to the differential equation though. Any of the subsequent is also solutions to the differential equation.

y (t ) = -9e^3t

y (t ) = 56e^-3t

y (t ) = 7e^3t  - 6^e-3t

 y (t ) = 123e^3t

y (t ) = (14/9) e^-3t

y (t )= -92e^3t  -16^e-3t

 Actually, if you think about it any function which is in the form

y (t ) = c e^3t  + c e^-3t will be a solution to the differential equation.

This illustration leads us to a very significant fact that we will use in practically each problem in this section will be a solution to the differential equation.