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Solving this equation: y{\,^{iv}} + 5y\,''' - y\,'' + 8y\,' - 3y = 0 get a characteristic equation whose polynomial of 4th graders can not be factored by any known method, is not even factored by 2 2nd degree polynomials with pairs of complex roots. $ {r} ^ {4} +5 \ {r} ^ {3} - {r} ^ {2} +8 \, r-3 = 0 $ then: how would you find the solution of this ODE?

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    @mathsolomon: It might be worth noting that you can express the solutions in terms of the roots (even if you cannot write down the roots); but you *do* need to test to see if the polynomial has multiple roots or not. If it has no multiple roots, then the solutions are just linear combinations of $e^{\rho_i t}$, where $\rho_1,\rho_2,\rho_3,\rho_4$ are the roots; but if it *has* multiple roots, the expression changes a bit. Luckily, one can test to see if the polynomial has multiple roots without having to find the roots.2011-07-26

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There is in fact a thoroughly unpleasant closed form formula for the roots of a quartic. It was discovered by Ferrari, not the one of automobile fame, in the sixteenth century.

But the existence of a closed form formula is irrelevant. If you have a linear differential equation with constant coefficients, of any order, say for simplicity with no multiple roots, there is a simple expression for the solutions of the DE in terms of the roots of a certain polynomial.

Ultimately, we may end up having to approximate these roots. That is a familiar situation. Even when we have a closed form solution, to get numbers out we often need to approximate.

Added: Note that any polynomial with real coefficients can be factored as a product of linear and/or quadratic polynomials with real coefficients. There may not be a simple expression for these coefficients in terms of the coefficients of the original polynomial. But the coefficients of the factors can be found to any desired degree of accuracy.

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Wolfram Alpha reports four roots, two real and two complex. It declines to give expressions in terms of radicals, just numeric answers.

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    You can get the expressions in terms of radicals [here](http://www.wolframalpha.com/input/?i=r^4%2B5r^3-r^2%2B8r-3%3D0) by clicking the "Exact Forms" buttons. They're stupid complicated though.2011-07-26