Second order differential equation

[doroulla]

hi. I can't figure out this question:

d2y/dx2 - 2 dy/dx - 3y = x

(i) find complementary function
(ii) find particular integral
(iii) using (i) and (ii) find the general solution
(iv) find the solution that satisfies the initial conditions:
y=2/9 at x=0 and dy/dx=-13/3 at x=0


i did:

m^2 - 2m - 3 = 0
(m-3)(m+1)=0
real and distinct solutions thus

y = Ae^(3x) + Be^(-x)

thus dy/dx = 3Ae^3x - Be^-x

d2y/dx2 = 9Ae^3x + Be^-x

now i have no idea how to continue. As i understood what i found above is the complementary function. I think. Thank you

 

[rock.freak667]

y = Ae^(3x) + Be^(-x)

Yes this would be the complementary solution.

What the particular integral for a polynomial?

 

[doroulla]

i think its something to do with the x on the right hand side but i don't know how to do it. Do i integrate that side?

I know when you have a y in the equation you integrate the constants infront of the y. But in this case i don't have this.

 

[HallsofIvy]

Try a specific integral of the form $Ax^2+ Bx+ C$. What must A, B, and C equal to satisfy the equation? Recall that the general solution to the entire equation is the general solution to the associated homogeneous equation (the "complementary solution") plus a specific integral.