Find the complementary function

[doroulla]

the question is to
find the complementary function, particular integral and general solution of:
d2y/dx2 - 2dy/dx - 3y= x

i did: m^2 -2m -3 =0
i get (m-3)(m+1)=0
so i have y=Ae^3 + Be^-1
i don't know how to continue from here.. Is this the complementary function?

 

[tiny-tim]

welcome to pf!

hi doroulla! welcome to pf!

(try using the X² icon just above the Reply box )

… so i have y=Ae^3 + Be^-1
i don't know how to continue from here.. Is this the complementary function?

the general solution to the complementary equation is y = Ae^3x + Be^-x

if y₁ and y₂ are any two solutions to the original equation, then y₁ - y₂ is a solution to the complementary equation

now you need to find one particular solution

the particular solution depends only on the RHS of the original equation, and usually has nothing to do with the LHS (so the particular solution usually has nothing to do with the complementary solution)

this is mostly intelligent guesswork

in this case, try a polynomial ​