General Solution for y''-18y'+80y=0 w/ Variable t

[beanryu]

Here is my question

Find the general solution to the differential equation

y''-18y'+80y=0

Express the solution in terms of the variable t. Give the smaller root first.

My problem is that I don't know what general solution mean?
what does it mean?

And where does the t come from?

 

[AKG]

Example:

y'' - 5y' + 6y = 0

So you want to find all functions y that satisfy this equation. If we say that y is a function of t, then y' denotes the derivative of y with respect to t. There's nothing special about t, however. A function y has to be a function of some variable. We could have very well called our variable, say, x. Then y' would denote the derivative of y with respect to x. Now the general solution to the differential equation I gave is:

y(t) = Ce^2t + De^3t

But here y depends not only on t, but also on C and D, so it might be better to write:

y_C,D(t) = Ce^2t + De^3t

What does it mean for y_C,D to be the general solution? It means two things:

1) for any choice of real numbers C and D, the resulting function is a solution to the differential equation. For example, the following two different functions:

y_7,12.9921(t) = 7e^2t + 12.9921e^3t
y_0,1(t) = e^3t

are both solutions to the differential equation. In order to check this, compute y_7,12.9921', y_7,12.9921'', and y_0,1', y_0,1''. Plug these into the differential equation, and see that in both cases, the left side really does end up being 0.

Again, the above is only two possible choices for the pair C,D. Any imaginable choice will do (normally, you restrict your choice of C and D to real numbers, but you could even choose complex numbers, quaternions, whatever!)

2) Every function which solves this differential equation is in the form Ce^2t + De^3t. That is, if f is some function of t, and there are no number C and D such that

f(t) = Ce^2t + De^3t

Then f can not be a solution to the differential equation. In other words, you will get every possible solution to the differential equation if you consider every possible choice of C and D in the general solution y_C,D(t) = Ce^2t + De^3t. That is y is called the general solution, because it characterizes every solution.

You must already have been taught how to find the general solution to such a differential equation.