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So I'm working with a nonhomogeneous second order differential equation:

$$4y''-y=\sin(x)\cos(x/2).$$

I know that the general solution, $y$, equals $y_c + y_p$ where $y_c$ is the general solution to the complementary equation and $y_p$ is any particular solution to the nonhomogeneous equation. I'm struggling a little bit with $y_p$ because I'm not sure what form the particular solution should be.

I know (at least I think I do) that, for example, the general form of the particular solution for $\cos(x/2)$ is:

$$A\sin(x/2) + B\cos(x/2).$$

I also suspect that the general form of the particular solution for $\sin(x) + \cos(x/2)$ is:

$$A\sin(x) + B\cos(x) + C\sin(x/2) + D\cos(x/2).$$

However, I'm completely thrown off track with $\sin(x)\cdot\cos(x/2)$. I'd appreciate any insight on the matter, because frankly, the entire concept is still a little loose in my head.

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    What do you mean by *the general form of the particular solution for cos(x/2)*? Usually, what is called *a solution* is a solution *of a diffeential equation* so you should explain what differential equation you are considering.2011-07-10
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    Was about to say the same thing... to speak of a solution of a differential equation, you actually need a differential equation. =P2011-07-10
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    hope my edit adds a little bit of clarity to my challenge2011-07-10
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    Wolfram Alpha can solve this for you: [calculation](http://www.wolframalpha.com/input/?i=4*y%27%27%28x%29-y%28x%29%3Dsin%28x%29*cos%28x%2F2%29) Now that you know the result can you think about it again and come up with the right solution?2011-07-10
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    It seems you simply need to learn the so-called *variation of parameters* method. It is competently explained here: http://en.wikipedia.org/wiki/Variation_of_parameters, see in particular *2.1 Specific second order equation* http://en.wikipedia.org/wiki/Variation_of_parameters#Specific_second_order_equation2011-07-10
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    @Richard: Are you studying mathematics on your own?2011-07-10
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    I think I'll look into the variation of parameters method, and I appreciate Wolfram Alpha program as well. Should make things a lot simpler for me. I'm actually taking a calculus course right and both the professor and textbook material have (I feel) been vary vague on solving nonhomogeneous second order differential equations. The vast majority of my learning has come from internet sources.2011-07-10
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    Try a solution of the form $A \sinh(x/2) + B \cosh(x/2) $ since it solves $4y''-y=0$.2011-07-11

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