What are the solutions and domains for Homogenous Linear Equations (Wave)?

[ineedhelpnow]

(Wave)
i have a test on friday that I am studying for so i was working through some problems in my textbook. i came across this question and I am stuck on what to do. can anyone help me out?
thanks
View attachment 4128

 

[MarkFL]

What you can do here is take the first and second derivatives (with respect to $x$) of the solutions given and substitute them into the given ODE and see if an identity results. For example, let's do $y_1$:

\(\displaystyle y_1=e^{2x}\)

And so we find:

\(\displaystyle y_1'=2e^{2x}\)

\(\displaystyle y_1''=4e^{2x}\)

And then substituting these into the ODE, we find:

\(\displaystyle 4e^{2x}-7\cdot2e^{2x}+10e^{2x}=0\)

\(\displaystyle 0=0\)

Thus, we know $y_1$ is a solution of (A). Try the other two...:D

 

[ineedhelpnow]

I understand now ^^ but what if it were on a different interval from like 0 to 1 instead. Would that make a difference or does it mean that the function only exists on this interval?

 

[MarkFL]

As given, the solutions are defined for all real $x$. The ODE and/or solution will tell you where the solution is defined, either explicitly, or implied. :D