Final answer:
The characteristic equations for the given ODEs are r^2+7r+3=0 and 4r^2+8r+13=0. These equations need to be solved to find the cases and values of r that satisfy them.
Step-by-step explanation:
The given equations are 2y′′+7y′+3y=0 and 4y′′+8y′+13y=0.
To find the characteristic equations, we assume a solution of the form y=e^rx, where r is a constant.
Substituting this into the first equation, we get r^2+7r+3=0, which gives us the characteristic equation r^2+7r+3=0.
Similarly, substituting the solution form into the second equation, we get 4r^2+8r+13=0, which gives us the characteristic equation 4r^2+8r+13=0.
The characteristic equations for the given ODEs are r^2+7r+3=0 and 4r^2+8r+13=0. These equations need to be solved to find the cases and values of r that satisfy them.