Question

By substitution, the function y(p)=-(11/12)-(1/2)x is readily shown to be a particular solution of the nonhomogeneous equation y'''-6y''+11y'-6y=3x. Find the general solution.

Answer 1

The general solution is the homogenous solution + the particular solution.
To find the homogeneous solution set it equal to zero giving:
\[y'''-6y''+11y'-6y=0\]
From here you know that, if you were to reduce it down to a first order system, that you can replace the y's with the eigenvalues (lambda) and the primes as powers. This gives:
\[\lambda^3-6\lambda^2+11\lambda-6=0\]
Then you need to factor this polynomial. It factors to:
\[(\lambda-2)(\lambda-3)(\lambda-1)\]
So you know you have 3 real roots, 1,2,3. So your homogeneous solution should be:
\[y_H(t)=c_1e^{t}+c_2e^{2t}+c_3e^{3t}\]
Adding this to your particular should give you the answer you seek.

Answer 2

And wolfram agrees: http://www.wolframalpha.com/input/?i=solve+y%27%27%27-6y%27%27%2B11y%27-6y%3D3x

Answer 3

Thank you!

Answer 4

No problem, the hardest part of higher order systems is factoring the characteristic polynomial.