Math Help - general solution to differential equation

1. general solution to differential equation

For our practice final one of the questions is

Find the general solution to the following differential equation: y'' - 4y' + 4y =x2 - 3x + 2

I understand that one would add the particular solution and general solution. For the particular solution I got 1/4x2-1/4x+1/8 and for the general solution I got Ae2x+Be2x. However he gave us the solution and it was Ae-2x+Bxe-2x+1/4x2-1/4x+1/8. Does anyone see where I went wrong? I can show more work if thats helpful.

2. Re: general solution to differential equation

Originally Posted by JML2618
For our practice final one of the questions is

Find the general solution to the following differential equation: y'' - 4y' + 4y =x2 - 3x + 2

I understand that one would add the particular solution and general solution. For the particular solution I got 1/4x2-1/4x+1/8 and for the general solution I got Ae2x+Be2x. However he gave us the solution and it was Ae-2x+Bxe-2x+1/4x2-1/4x+1/8. Does anyone see where I went wrong? I can show more work if thats helpful.
Your mistake was in not dealing with the fact that your characteristic polynomial has a single root of order 2.
That makes your homogeneous solution of the form

$y[x]=c_1e^{2x}+c_2xe^{2x}$ (note the factor x in the 2nd term)

go use this form for your homogeneous solution and resolve your particular solution.

3. Re: general solution to differential equation

oh okay i understand now thanks for the reply! Do you use this whenever it's in the form y''+y'+y and ce^ax +ce^ax when it's y''+y'?

4. Re: general solution to differential equation

or is it that the roots are equal?

5. Re: general solution to differential equation

Originally Posted by JML2618
or is it that the roots are equal?
Pauls Online Notes : Differential Equations - Repeated Roots

6. Re: general solution to differential equation

Notice that $Ae^{2x}+ Be^{2x}= Ce^{2x}$ with C= A+ B. Those are not two independent solutions and cannot give the general solution to a second order equation.