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3.5:  Solving non-homogeneous linear DE  using the undetermined coefficients method
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1.)  Step 1:  Solve homogeneous version of DE.
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2.)  Step 2:  Guess a non-homogeneous solution with undetermined coefficients. Plug into the non-homogeneous linear DE to solve for the undetermined coefficients.
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3.)  Combing general homogeneous solution with a non-homogeneous solution. 

{\bf Starting guess:}
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\t If $ay'' + by' + cy = ke^{pt}$,  guess $y = Ae^{pt}$

\t If $ay'' + by' + cy = ksin(pt) + j cos(pt)$,  
{guess $y = Asin(pt) + Bcos(pt)$}

\t If $ay'' + by' + cy = $ degree $n$ polynomial,  

\rightline{guess $y =$ a degree $n$ polynomial including all terms}

\rightline{ (with undetermined coefficients) including constant term.}

\t If $ay'' + by' + cy = $ a sum, guess a sum (but usually solve separately).

\t If $ay'' + by' + cy = $ a product, guess a product.

{\bf Sometimes the above can be simplified:}
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\t  If a term does not show up when you take the derivatives of $y$, you may be able to omit that term.  E.g,  $y'' + w^2y = sin(pt)$ where $p \not= w$, then $y = Asin(pt)$ is a simpler guess that works.

{\bf If the above does not work}
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\t  Try multiplying non-simplified guess by $t$.

 Example:  If guess is a homogeneous solution, then that will not be a non-homogeneous solution.  Thus must guess something else.  Multiplying  non-simplified guess by $t$ until no longer homogeneous works.  

 Example: If $y$ term missing, and $g(t) =$ degree $n$ polynomial,  then will need to multiply by $t$ so that when you plug in guess, you will have a degree $n$ polynomial on both sides of equal sign.

Note:  you are  multiplying the {\bf non-simplified guess} by $t$.  When you take derivatives of $y$, you must use the {\bf product} rule.  Thus extra terms appear when you take the derivative and you will need the non-simplified guess to cancel out these terms.




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