EqWorld http://eqworld.ipmnet.ru Exact Solutions > Ordinary Differential Equations > Higher-Order Linear Ordinary Differential Equations > Constant Coefficient Linear Differential Equation 15. a n y (n) x + a n–1 y (n–1) x + ··· + a 1 y 0 x + a 0 y = 0, a n ≠ 0. Constant coefficient linear homogeneous differential equation. The general solution of this equation is determined by the roots of the characteristic equation: P (λ) =0, where P (λ) = a n λ n + a n-1 λ n-1 + ··· + a 1 λ + a 0 . The following cases are possible: 1 ◦ . All roots λ 1 , λ 2 , ... , λ n of the characteristic equation are real and distinct. Then the general solution of the homogeneous linear differential equation (1) has the form: y = C 1 exp(λ 1 x) + C 2 exp(λ 2 x) + ··· + C n exp(λ n x). 2 ◦ . There are m equal real roots λ 1 = λ 2 = ··· = λ m (m ≤ n), and the other roots are real and distinct. In this case, the general solution is given by: y = exp(λ 1 x)(C 1 + C 2 x + ··· + C m x m-1 ) + C m+1 exp(λ m+1 x) + C m+2 exp(λ m+2 x) + ··· + C n exp(λ n x). 3 ◦ . There are m equal complex conjugate roots λ = α iβ (2m ≤ n), and the other roots are real and distinct. In this case, the general solution is: y = exp(αx) cos(βx)(A 1 + A 2 x + ··· + A m x m-1 ) + exp(αx) sin(βx)(B 1 + B 2 x + ··· + B m x m-1 ) + C 2m+1 exp(λ 2m+1 x) + C 2m+2 exp(λ 2m+2 x) + ··· + C n exp(λ n x), where A 1 , ... , A m , B 1 , ... , B m , C 2m+1 , ... , C n are arbitrary constants. 4 ◦ . In the general case, where there are r different roots λ 1 , λ 2 , ... , λ r of multiplicities m 1 , m 2 , ... , m r , respectively, the right-hand side of the characteristic equation can be represented as the product P (λ) = (λ - λ 1 ) m 1 (λ - λ 2 ) m 2 ... (λ - λ r ) mr , where m 1 + m 2 + ··· + m r = n. The general solution of the original equation is given by the formula: y = r X k=1 exp(λ k x)(C k,0 + C k,1 x + ··· + C k,m k -1 x m k -1 ), where C k,l are arbitrary constants. If the characteristic equation has complex conjugate roots, then in the above solution, one should extract the real part on the basis of the relation exp(α iβ) = e α (cos β i sin β). References Kamke, E., Differentialgleichungen: L¨ osungsmethoden und L¨ osungen, I, Gew¨ ohnliche Differentialgleichungen, B. G. Teubner, Leipzig, 1977. Boyce, W. E. and DiPrima, R. C., Elementary Differential Equations, 7th Edition, Wiley, New York, 2000. Polyanin, A. D. and Zaitsev, V. F., Handbook of Exact Solutions for Ordinary Differential Equations, 2nd Edition , Chapman & Hall/CRC, Boca Raton, 2003. Constant Coefficient Linear Differential Equation Copyright c 2004 Andrei D. Polyanin http://eqworld.ipmnet.ru/en/solutions/ode/ode0415.pdf

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Exact Solutions > Ordinary Differential Equations > Higher-Order Linear Ordinary Differential Equations >Constant Coefficient Linear Differential Equation

15. any(n)x + an–1y

(n–1)x + · · · + a1y

′x + a0y = 0, an ≠ 0.

Constant coefficient linear homogeneous differential equation.The general solution of thisequation is determined by the roots of the characteristic equation:

P (λ) = 0, where P (λ) = anλn + an−1λn−1 + · · · + a1λ + a0.

The following cases are possible:

1. All roots λ1, λ2, . . . , λn of the characteristic equation are real and distinct. Then the generalsolution of the homogeneous linear differential equation (1) has the form:

y = C1 exp(λ1x) + C2 exp(λ2x) + · · · + Cn exp(λnx).

2. There arem equal real rootsλ1 = λ2 = · · · = λm (m ≤ n), and the other roots are real anddistinct. In this case, the general solution is given by:

y = exp(λ1x)(C1 + C2x + · · · + Cmxm−1)

+ Cm+1 exp(λm+1x) + Cm+2 exp(λm+2x) + · · · + Cn exp(λnx).

3. There arem equal complex conjugate rootsλ = α ± iβ (2m ≤ n), and the other roots are realand distinct. In this case, the general solution is:

y = exp(αx) cos(βx)(A1 + A2x + · · · + Amxm−1)

+ exp(αx) sin(βx)(B1 + B2x + · · · + Bmxm−1)

+ C2m+1 exp(λ2m+1x) + C2m+2 exp(λ2m+2x) + · · · + Cn exp(λnx),

whereA1, . . . , Am, B1, . . . , Bm, C2m+1, . . . , Cn are arbitrary constants.

4. In the general case, where there arer different roots λ1, λ2, . . . , λr of multiplicitiesm1, m2, . . . , mr, respectively, the right-hand side of the characteristic equation can be representedas the product

P (λ) = (λ − λ1)m1 (λ − λ2)m2 . . . (λ − λr)mr ,

wherem1 +m2 + · · ·+mr = n. The general solution of the original equation is given by the formula:

y =r∑

k=1

exp(λkx)(Ck,0 + Ck,1x + · · · + Ck,mk−1xmk−1),

whereCk,l are arbitrary constants.If the characteristic equation has complex conjugate roots, then in the above solution, one should

extract the real part on the basis of the relation exp(α ± iβ) = eα(cosβ ± i sinβ).

ReferencesKamke, E., Differentialgleichungen: Losungsmethoden und Losungen, I, Gewohnliche Differentialgleichungen, B. G.

Teubner, Leipzig, 1977.Boyce, W. E. and DiPrima, R. C.,Elementary Differential Equations, 7th Edition, Wiley, New York, 2000.Polyanin, A. D. and Zaitsev, V. F.,Handbook of Exact Solutions for Ordinary Differential Equations, 2nd Edition, Chapman

& Hall/CRC, Boca Raton, 2003.

Constant Coefficient Linear Differential Equation

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