Examples: Solving nth Order Equations
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1 Atoms L. Euler s Theorem The Atom List First Order. Solve 2y + 5y = 0. Examples: Solving nth Order Equations Second Order. Solve y + 2y + y = 0, y + 3y + 2y = 0 and y + 2y + 5y = 0. Third Order. Solve y y = 0 and y y = 0. Fourth Order. Solve y iv y = 0.
2 Atoms An atom is a term with coefficient 1 obtained by taking the real and imaginary parts of x j e ax (cos cx + i sin cx), j = 0, 1, 2,..., where a and c represent real numbers and c 0. By definition, zero is not an atom. Theorem 1 (L. Euler) The function y = x j e r 1x is a solution of a constant-coefficient linear homogeneous differential of the nth order if and only if (r r 1 ) j+1 divides the characteristic polynomial. Euler s theorem is used to construct solutions of the nth order differential equation. The solutions so constructed are n distinct atoms, hence independent. Picard s theorem implies the list of atoms is a basis for the solution space.
3 The Atom List 1. If r 1 is a real root, then the atom list for r 1 begins with e r 1x. The revised atom list is e r 1x, xe r 1x,..., x k 1 e r 1x provided r 1 is a root of multiplicity k. This means that factor (r r 1 ) k divides the characteristic polynomial, but factor (r r 1 ) k+1 does not. 2. If r 1 = α + iβ, with β > 0 and its conjugate r 2 = α iβ are roots of the characteristic equation, then the atom list for this pair of roots (both r 1 and r 2 counted) begins with e αx cos βx, e αx sin βx. For a root of multiplicity k, these real atoms are multiplied by atoms 1,..., x k 1 to obtain a list of 2k atoms e αx cos βx, xe αx cos βx,..., x k 1 e αx cos βx, e αx sin βx, xe αx sin βx,..., x k 1 e αx sin βx.
4 1 Example (First Order) Solve 2y + 5y = 0 by Euler s method, verifying y h = c 1 e 5x/2. 2y + 5y = 0 Given differential equation. 2r + 5 = 0 Characteristic equation. Find it by replacement y (n) r n. r = 5/2 Exactly one real root. Atom = e 5x/2 For a real root r, the atom is e rx. y h = c 1 e 5x/2 The general solution y h is written by multiplying the atom list by constants c 1, c 2,....
5 2 Example (Second Order I) Solve y + 2y + y = 0 by Euler s method, showing y h = c 1 e x + c 2 xe x. y + 2y + y = 0 Given differential equation. r 2 + 2r + 1 = 0 Characteristic equation. Use y (n) r n. r = 1, 1 Exactly two real roots. Atoms = e x, xe x For a double root r, the atom list is e rx, xe rx. y h = c 1 e x + c 2 xe x The general solution y h is written by multiplying the atom list by constants c 1, c 2,....
6 3 Example (Second Order II) Solve y +3y +2y = 0 by Euler s method, showing y h = c 1 e x + c 2 e 2x. y + 3y + 2y = 0 Given differential equation. r 2 + 3r + 2 = 0 Characteristic equation. Use y (n) r n. r = 1, 2 Factorization (r + 2)(r + 1) = 0. Atoms = e x, e 2x For a real root r of multiplicity one, the atom is e rx. y h = c 1 e x + c 2 e 2x The general solution y h is written by multiplying the atom list by constants c 1, c 2,....
7 4 Example (Second Order III) Solve y +2y +5y = 0 by Euler s method, showing y h = c 1 e x cos 2x + c 2 xe x sin 2x. y + 2y + 5y = 0 Given differential equation. r 2 + 2r + 5 = 0 Characteristic equation. Use y (n) r n. r = 1 + 2i, 1 2i Factorization (r + 1) = 0. Atoms = e x cos 2x, e x sin 2x For a complex root r = α + iβ of multiplicity one, the atoms are e αx cos βx and e αx sin βx. y h = c 1 e x cos 2x + c 2 e x sin 2x The general solution y h is written by multiplying the atom list by constants c 1, c 2,....
8 5 Example (Third Order I) Solve y y = 0 by Euler s method, showing y h = c 1 + c 2 e x + c 3 e x. y y = 0 Given differential equation. r 3 r = 0 Characteristic equation. Use y (n) r n. r = 0, 1, 1 Factorization r(r + 1)(r 1) = 0. Atoms = 1, e x, e x y h = c 1 + c 2 e x + c 3 e x For a real root r of multiplicity one, the atom is e rx. The general solution y h is written by multiplying the atom list by constants c 1, c 2, c 3,....
9 6 Example (Third Order II) Solve y y = 0 by Euler s method, showing y h = c 1 + c 2 x + c 3 e x. y y = 0 Given differential equation. r 3 r 2 = 0 Characteristic equation. Use y (n) r n. r = 0, 0, 1 Factorization r 2 (r 1) = 0. Atoms = 1, x, e x For a real root r of multiplicity one, the atom is e rx. y h = c 1 + c 2 x + c 3 e x The general solution y h is written by multiplying the atom list by constants c 1, c 2, c 3,....
10 7 Example (Fourth Order) Solve y iv y = 0 by Euler s method, showing y h = c 1 + c 2 x + c 3 e x + c 4 e x. y iv y = 0 Given differential equation. r 4 r 2 = 0 Characteristic equation. Use y (n) r n. r = 0, 0, 1, 1 Factorization r 2 (r 1)(r + 1) = 0. Atoms = 1, x, e x, e x y h = c 1 + c 2 x + c 3 e x + c 4 e x For a real root r of multiplicity one, the atom is e rx. For a double root, the atoms are e rx, xe rx. The general solution y h is written by multiplying the atom list by constants c 1, c 2, c 3, c 4,....
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