Constructing a differential equation from the solution

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To construct a differential equation from the given solution y = c1e3x + c2xe3x + c3e2xsin(x) + c4e2xcos(x), the roots k = 3 (with multiplicity 2) and k = 2 ± i must be incorporated. The characteristic polynomial can be formed using these roots, leading to a fourth-order differential equation. The relationship between the roots and the degree of the derivatives indicates that the order of the differential equation corresponds to the number of roots, including their multiplicities. Understanding how to combine the real and complex roots is crucial for deriving the correct differential equation. The final equation will reflect the structure of the solution provided.
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Homework Statement



y = c1e3x+c2xe3x+c3e2xsin(x)+c4e2xcos(x)

Homework Equations



Differential Equations.

The Attempt at a Solution



I have the roots of k=3, k=3, k=2+i, k=2-i.
Now I am just stuck on how to put the roots together to get the original equation. I am just stuck on the complex number part.
 
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What equation is it which has the roots 3, 3, 2+i and 2-i? How do you get such equation from the differential equation? How the powers of k and the degree of the derivatives are related?

ehild
 

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