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?
How do I solve the ODE $y^{iv} + 5y'''- y''+ 8y' - 3y = 0 $?
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0@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
2 Answers
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.
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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1You 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