show that every member of the family of functions y = aexp^r1x + bexp^r2x is also a solution.

Question

The differential equation was 6y''+3y'-3y=0

For what values of r does the function y = e^rx satisfy the equation?

r=-1

r=1/2

I get to plug in you take the derivative but I dont get the part when it asks show that every member of the family of functions.

Answer 1

show that every member of the family of functions:

This means that every member of this family of functions, for example

2e^-x+3e^(1/2)x

or

3e^-x+5e^(1/2)x

or

any similar functions but different constant

are the solution of this differential equation

Thus, we have to prove this by differentiating the functions (ae^-x + be^(1/2)x) and substitute it in the equation to show that any constant a and b are feasible