Undetermined Coefficients

• Mar 6th 2011, 10:49 AM
JJ007
Undetermined Coefficients
Set up the correct linear combination of functions with undetermined literal coefficients to use in finding a particular integral by the method of UC. (Do not actually find the particular integrals.)

\$\displaystyle y''-6y'+8y = x^3+x+e^{-2x}\$

\$\displaystyle m^2-6m+8=0\$

\$\displaystyle y_c=C_1e^{2x}+C_2e^{4x}\$

\$\displaystyle S_1=(x^3,x^2,x,1)\$
\$\displaystyle S_2=(x,1)\$
\$\displaystyle S_3=(e^{-2x})\$

S_2 is completely contained in S_1 but none are included in the complimentary function?

Thanks.
• Mar 6th 2011, 11:39 AM
TheEmptySet
Quote:

Originally Posted by JJ007
Set up the correct linear combination of functions with undetermined literal coefficients to use in finding a particular integral by the method of UC. (Do not actually find the particular integrals.)

\$\displaystyle y''+6y'+8y = x^3+x+e^{-2x}\$

\$\displaystyle m^2-6m+8=0\$

\$\displaystyle y_c=C_1e^{2x}+C_2e^{4x}\$

\$\displaystyle S_1=(x^3,x^2,x,1)\$
\$\displaystyle S_2=(x,1)\$
\$\displaystyle S_3=(e^{-2x})\$

S_2 is completely contained in S_1 but none are included in the complimentary function?

Thanks.

You have a typo in your characteristic equation. It should read

\$\displaystyle m^2+6m+8=(m+2)(m+4)\$
• Mar 6th 2011, 11:58 AM
JJ007
Quote:

Originally Posted by TheEmptySet
You have a typo in your characteristic equation. It should read

\$\displaystyle m^2+6m+8=(m+2)(m+4)\$

Sorry. The typo was in the original equation. So it's still (m-2)(m-4).
• Mar 6th 2011, 12:03 PM
TheEmptySet
Since the complimentary solution does not appear in the right hand side you can just "guess" the most simple form

\$\displaystyle y=Ax^3+Bx^2+Cx+D+Me^{-2x}\$