For what values of r does the function y = e^rx satisfy the equation y'' + 4y' - 45y = 0?
Hello,
Why don't you just try to plug y = e^rx into the given equation and see what happens? You should notice that e^rx is never equal to zero, so this will lead you to a polinomial equation in r. Solve it and find the roots - they are what you're looking for.
An eqn in the form $\displaystyle y''+ay'+by=0$ has a characteristic eqn $\displaystyle r^2+ar+b=0$ which gives the solution to the original D.E for real values of $\displaystyle r$ in the form $\displaystyle y = e^{rx}$
Did you solve for $\displaystyle r$ ? What did you get?