Two Degree non Homogeneous Differential Equations with Polynomial Coefficients

Transcription

1 Mathematica Aeterna, Vol. 1, 2011, no. 05, Two Degree non Homogeneous Differential Equations with Polynomial Coefficients Nikos Bagis Aristotelian University of Thessaloniki Department of Computer Sciences Thessaloniki Greece Abstract We solve some forms of non homogeneous differential equations using a new function u g which is integral-closed form solution of a non homogeneous second order ODE with linear coefficients. The non homogeneous part is an arbitrary function of L 2 (R). Using this function u g as a base we give in closed integral form solutions of several two degree DE. Mathematics Subject Classification: 34A05, 34A25, 34A35 Keywords: Differential equations, Linear, Non homogeneous, Closed form, Solutions, Fourier transforms 1 Introduction We will solve the equation (a 1 x+b 1 )f (x)+(a 2 x+b 2 )f (x)+(a 3 x+b 3 )f(x) = g(x) (1) where f, g L 2 (R) and a 1, a 2, a 3, b 1, b 2, b 3 are constants in R. We call the solution u g, and using this solution we try to solve other general differential equations. The solution of (1) (function u g ), follow using the Fourier transform and its properties. The result is a quite complicated integral. However the Fourier transform of this integral is not so complicated and satisfies as we will see a simple first order differential equation. We also consider and solve completely a class of differential equations having two degree polynomial coefficients. We also give examples and applications. Special cases of u g are known functions, such as Hermite polynomials, Bessel functions, confluent hypergeometric functions and other.

2 298 Nikos Bagis 2 Preliminary Notes Let the Fourier Transform of a function f of L 2 (R) is f(γ) = the Inverse Fourier Transform is f(x) = 1 2π Then it is known (integration by parts) (see [2]). = +2( iγ) f(t)e itγ dx f(γ)e iγx dγ f(x)x n e ixγ dx = i n ( f) (n) (γ). (2) (f (n) )(γ) = (iγ) n f(γ). (3) f (x)a(x)e ixγ dx = f(x)a (x)e ixγ dx+( iγ) 3 Main Results f (x)a(x)e ixγ = f(x)a (x)e ixγ dx+( iγ) 2 These are the main results of the paper. f(x)a(x)e ixγ dx. (4) f(x)a (x)e ixγ dx+ f(x)a(x)e ixγ dx. (5) Theorem 3.1 When f, g L 2 (R) and lim x f(x)x 2+ǫ = 0, ǫ > 0, equation (1) can reduced to ( ia 1 γ 2 +a 2 γ +ia 3 ) which is solvable. f(γ) dγ +( b 1γ 2 2ia 1 γ +ib 2 γ +a 2 +b 3 ) f(γ) = ĝ(γ) (6) Proof. Take the Fourier Transform in both sides of(1) and use relations(2),(3),(4),(5). We immediately arrive to(6). But (6) is completely solvable DE of first degree.

3 Two Degree non Homogeneous Differential Equations 299 After solving it we use the inverse Fourier transform to get the solution, which we call it f(x) = u g [{a 1,a 2,a 3 },{b 1,b 2,b 3 };x]. (7) and satisfies (6). Note. When g(x) = 0, the solution is very simple as someone can see is û 0 [{a 1,a 2,a 3 },{b 1,b 2,b 3 };γ] = ( b1 γ 2 ) 2ia 1 γ +ib 2 γ +a 2 +b 3 = exp dγ ia 1 γ 2 +a 2 γ +ia 3 (8) Theorem 3.2 The DE (ax 2 +bx+c)y (x)+(kx 2 +lx+m)y (x)+(tx 2 +rx+s)y(x) = g(x) (9) is solvable if k = a(4a bl l +2am) b 2 +2ac b (10) s = 6a 2b2 c + 2b l+ bm c c m c and = b 2 4ac The solution is + br 2a + r 2a b2 t 2a 2 + ct a b t 2a 2 (11) y(x) = a 1x+b 1 ax 2 +bx+c u g[{a 1,a 2,a 3 },{b 1,b 2,b 3 };x] (12) where a 1 = a 2 = a 3 = b 1 = ab 3 (b+ ) (b+ )k +2a(2a l+s) b 3 (b+ )k (b+ )k +2a(2a l+s) b 3 (b+ )t (b+ )k +2a(2a l+s) 2ab 3 c (b+ )k +2a(2a l+s) b 2 = b 3(b+ )(4a 2 +bk + k 2al) 2a(4a 2 +bk + k 2al +2as) (13) (14) (15) (16) (17)

4 300 Nikos Bagis Proof. We proceed as in Proposition 1. Set y(x) = a 1x+b 1 Y(x) (18) ax 2 +bx+c We arrive in a equation (a 1 x+b 1 )Y (x)+(a 2 x+b 2 )Y (x)+(a 3 x+b 3 )Y(x) = g(x), (19) in which the relation of the coefficients of the two equations (9) and (19) is: b = ab2 1 +a2 1 c a 1 b 1, k = aa 2 a 1, l = 2a+ ab 2 a 1 + a 2c b 1, t = aa 3 a 1 m = 2ab2 1 +a 1b 2 c a 1 b 1, r = aa 2b 1 +ab 1 b 3 +a 1 a 3 c a 1 b 1, s = ab 1b 2 +a 1 b 3 c a 1 b 1 Solving the above system we get the proof. i) Theorem 2 prove that, a two degree ODE with second degree polynomial coefficients is 6-th parameter solvable DE and admits a general solution y(x) = u g (x) in a closed integral form. Thus if we give any values we want in (9) then it is solvable always except for two values of the coefficients (here) k and s, which are determined by the others. ii) A way to find some special cases of solutions of the DE (9) is with the command SolveAlways of the program Mathematica. Example 1. The equation ( x 2 +3x+1/3 ) y (x)+ +y(x) ( x 2 2 ( x x+3 ) y (x)+ ) x+ = 0 (20) 12 is equivalent to the second order, first degree polynomial coefficients ODE ( (9+ 69)x)Y(x)+2(2 69+( )x)Y (x)+ which is solvable according to Theorem 1. y(x) = +(2+(9+ 69)x)Y (x) = 0, (21) 9(2+(9+ 69)x) ( )(1+9x+3x 2 ) u 0 [{ 6+ 69,2( ), 9 69 }, { 2,4 69, }] where u 0 is given by (8).

5 Two Degree non Homogeneous Differential Equations 301 Theorem 3.3 The DE [ ax 2 + ab ] [ 1 x y (x)+ kx 2 +(2a+s)x+ 2ab ] 1 y (x)+ [ tx 2 +rx+s ] y(x) = g(x), a 1 a 1 (22) with g L 2 (R), have a closed integral form solution y(x) = u g [{ a 1, a 1k a, a } 1t, a { b 1, a 1s a, a } 1k +a 1 r a ;x ] (23) Applications in the case of linear coefficients 1. The homogeneous of the differential equation 2xy (x)+y (x) 2y(x) = g(x) (24) is and have solutions 2xy (x)+y (x) 2y(x) = 0 (25) With our method (24) have solution where y 1 (x) = cosh(2 x), y 2 (x) = sinh(2 x) y(x) = u g [{2,0,0},{1,1, 2};x] ŷ(w) = e i/w w 3/2 ( C 1 + w and G(x) = ĝ(x). 2. The homogeneous of the differential equation c ie i/x ) G(x) 2 dx x y (x) 2xy (x)+2ny(x) = g(x) (26) is and have solution y (x) 2xy (x)+2ny(x) = 0 (27) y 1 (x) = H n (x), y 2 (x) = 1 F 1 [ n 2, 1 2 ;x2 ]

6 302 Nikos Bagis With our method (26) have solution y(x) = u g [{0, 2,0},{1,0,2n};x] where ŷ(w) = e i(1+n)arctan( 2 w) w 1+n ( 4+w 2) 1 2 n 2 [C 1 + w + ie i(1+n)arctan( x) 2 G(x)x n (2i+x) ( 4+x 2)1 2 ( 1+n) dx] c and G(x) = ĝ(x). 3. The homogeneous of the differential equation xy (x)+y(x) = g(x) (28) is and have solution xy (x)+y(x) = 0 (29) y(x) = x [ C 1 J 1 (2 x)+c 2 Y 1 (2 x) ] With our method (28) have solution where y(x) = u g [{1,0,0},{0,0,1};x] ŷ(w) = 1 w 2e i w ( w ) C 1 + ie i x G(x)dx c and G(x) = ĝ(x). Note. In the above examples if we extract the general solutions and then set in the equations G(x) = 0, we arrive to the reduced homogeneous solution. Applications for the two degree polynomials coefficients 1. The homogeneous of the differential equation x 2 y (x)+[kx 2 +x(2+s)]y (x)+(tx 2 +rx+s)y(x) = g(x) (30) is x 2 y (x)+[kx 2 +x(2+s)]y (x)+(tx 2 +rx+s)y(x) = 0 (31)

7 Two Degree non Homogeneous Differential Equations 303 and have solution y(x) = = C ( 1 k 2r +k(2+s)+s x e kx/2 2 4tx/2 k2 4t U 2,s,x ) k 2 4t + k 2 4t + C 2 k 2r k(2+s) s L( x e kx/2 2 4tx/2 k2 4t 2, 1+s,x ) k 2 4t k 2 4t WhereL a n(x) = L(n,a,x) isthegeneralized Laguerrepolynomial andu(a,b,x) is confluent hypergeometric function. With our method (30) have solution 2. The equation y(x) = a 1 x u g[{a 1,a 1 k,a 1 t},{0,a 1 s, a 1 k +a 1 r};x] x(1 x)y (x)+(kx 2 +lx 2 k l)y (x)+[tx 2 +( s+2+2k+l t)x+s]y(x) = g(x) have solution y(x) = a 1 1 x u g [{a 1, a 1 k, a 1 t},{0, 2a 1 a 1 k a 1 l, 2a 1 a 1 k a 1 l+a 1 s};x] Example of the above equation we get if we choose b 1 = 0, a 2 = 0, b 2 = 2a 1 a 1 l, a 3 = a 1 t, b 3 = 2a 1 a 1 l+a 1 s and k = 0, then w ŷ(w) = e i(2+l s) w w 4 l C 1 + c ie i(2+l s) x G(x)x 2+l dx a 1 or is the solution of y(x) = a 1 1 x u g[{a 1,a 2,a 3 },{b 1,b 2,b 3 };x] (1 x)xy (x)+(lx 2 l)y (x)+[tx 2 +(2+l t)x+s]y(x) = g(x) 3. The equation x(1 x)y (x)+(kx 2 +lx+m)y (x)+(tx 2 +rx+s)y(x) = g(x) (32) is solvable with the u g functions when k +l+m+2 = 0.

8 304 Nikos Bagis References [1] N.N. Lebedev, Special Functions and their Applications, Dover Publications, Inc. New York (1972) [2] A. Papoulis, The Fourier Integral and its Applications, McGraw-Hill Publications, New York (1974) [3] Murray R. Spiegel, Schaum s Outline of Theory and Problems of Fourier Analysis with Applications to Boundary Value Problems, McGraw-Hill, Inc, (1974). [4] Greek Graduate books in Differential Equations Received: July, 2011