Handbook of mathematics for engineers and scienteists part 207 pps

1410 FUNCTIONAL EQUATIONS TABLE T12.1 Particular solutions of the nonhomogeneous functional equation y(x + 1)–y(x)=f(x) No. Right-hand side of equation, f(x) Particular solution, ¯y(x) 1 1 x 2 x 1 2 x(x – 1) 3 x 2 1 6 x(x – 1)(2x – 1) 4 x n , n = 0, 1, 2, 1 n+1 B n (x), where the B n (x) are Bernoulli polynomials. The generating function: te xt e t – 1 = ∞  n=0 B n (x) t n n! 5 1 x ψ(x)=–C +  1 0 1 – t x–1 1 – t dt is the logarithmic derivative of the gamma function, C = 0.5772 is the Euler constant 6 1 x(x + 1) – 1 x 7 a λx , a ≠ 1, λ ≠ 0 1 a λ – 1 a λx 8 sinh(a + 2bx), b > 0 cosh(a – b + 2bx) 2 sinhb 9 cosh(a + 2bx), b > 0 sinh(a – b + 2bx) 2 sinhb 10 xa x , a ≠ 1 1 a – 1 a x  x – a a – 1  11 ln x, x > 0 ln Γ(x), where Γ(x)=  ∞ 0 t x–1 e –t dt is the gamma function 12 sin(2ax), a ≠ πn – cos[a(2x – 1)] 2 sina 13 sin(2πnx) x sin(2πnx) 14 cos(2ax), a ≠ πn sin[a(2x – 1)] 2 sina 15 cos(2πnx) x cos(2πnx) 16 sin 2 (ax), a ≠ πn x 2 – sin[a(2x – 1)] 4 sina 17 sin 2 (πnx) x sin 2 (πnx) 18 cos 2 (ax), a ≠ πn x 2 + sin[a(2x – 1)] 4 sina 19 cos 2 (πnx) x cos 2 (πnx) 20 x sin(2ax), a ≠ πn sin(2ax) 4 sin 2 a – x cos[a(2x – 1)] 2 sina 21 x sin(2πnx) 1 2 x(x – 1)sin(2πnx) 22 x cos(2ax), a ≠ πn cos(2ax) 4 sin 2 a + x sin[a(2x – 1)] 2 sina 23 x cos(2πnx) 1 2 x(x – 1)cos(2πnx) 24 a x sin(bx), a > 0, a ≠ 1 a x a sin[b(x – 1)] – sin(bx) a 2 – 2a cos b + 1 25 a x cos(bx), a > 0, a ≠ 1 a x a cos[b(x – 1)] – cos(bx) a 2 – 2a cos b + 1 T12.1. LINEAR FUNCTIONAL EQUATIONS IN ONE INDEPENDENT VARIABLE 1411 2 ◦ . Solution for a < 0: y(x)=Θ 1 (x)|a| x sin(πx)+Θ 2 (x)|a| x cos(πx), where the Θ k (x)=Θ k (x + 1) are arbitrary periodic functions with period 1 (k = 1, 2). Remark. See also Subsection 17.2.1, Example 1. 4. y(x +1) – ay(x) = f(x). A nonhomogeneous first-order constant-coefficient linear difference equation. 1 ◦ . Solution: y(x)=  Θ(x)a x + ¯y(x)ifa > 0, Θ 1 (x)|a| x sin(πx)+Θ 2 (x)|a| x cos(πx)+¯y(x)ifa < 0, where Θ(x), Θ 1 (x), and Θ 2 (x) are arbitrary periodic functions with period 1, and ¯y(x)is any particular solution of the nonhomogeneous equation. 2 ◦ .Forf(x)= n  k=0 b k x n and a ≠ 1, the nonhomogeneous equation has a particular solution of the form ¯y(x)= n  k=0 A k x n ; the constants B k are found by the method of undetermined coefficients. 3 ◦ .Forf(x)= n  k=1 b k e λ k x , the nonhomogeneous equation has a particular solution of the form ¯y(x)= ⎧ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎩ n  k=1 b k e λ k – a e λ k x if a ≠ e λ m , b m xe λ m (x–1) + n  k=1, k≠m b k e λ k – a e λ k x if a = e λ m , where m = 1, , n. 4 ◦ .Forf(x)= n  k=1 b k cos(β k x), the nonhomogeneous equation has a particular solution of the form ¯y(x)= n  k=1 b k a 2 + 1 – 2a cos β k  (cos β k – a)cos(β k x)+sinβ k sin(β k x)  . 5 ◦ .Forf(x)= n  k=1 b k sin(β k x), the nonhomogeneous equation has a particular solution of the form ¯y(x)= n  k=1 b k a 2 + 1 – 2a cos β k  (cos β k – a)sin(β k x)–sinβ k cos(β k x)  . 1412 FUNCTIONAL EQUATIONS 5. y(x +1) – xy(x) =0. Solution: y(x)=Θ(x)Γ(x), Γ(x)=  ∞ 0 t x–1 e –t dt, where Γ(x) is the gamma function, Θ(x)=Θ(x + 1) is an arbitrary periodic function with period 1. The simplest particular solution corresponds to Θ(x) ≡ 1. 6. y(x +1) – a(x – b)(x – c)y(x) =0. Solution: y(x)=Θ(x)a x Γ(x – b)Γ(x – c), where Γ(x) is the gamma function, Θ(x) is an arbitrary periodic function with period 1. 7. y(x +1) – R(x)y(x) =0, R(x) = a (x – λ 1 )(x – λ 2 ) (x – λ n ) (x – μ 1 )(x – μ 2 ) (x – μ m ) . Solution: y(x)=Θ(x)a x Γ(x – λ 1 )Γ(x – λ 2 ) Γ(x – λ n ) Γ(x – μ 1 )Γ(x – μ 2 ) Γ(x – μ m ) , where Γ(x) is the gamma function, Θ(x) is an arbitrary periodic function with period 1. 8. y(x +1) – ae λx y(x) =0. Solution: y(x)=Θ(x)a x exp  1 2 λx 2 – 1 2 λx  , where Θ(x) is an arbitrary periodic function with period 1. 9. y(x +1) – ae μx 2 +λx y(x) =0. Solution: y(x)=Θ(x)a x exp  1 3 μx 3 + 1 2 (λ – μ)x 2 + 1 6 (μ – 3λ)x  , where Θ(x) is an arbitrary periodic function with period 1. 10. y(x +1) – f(x)y(x) =0. Here, f(x)=f (x + 1) is a given periodic function with period 1. Solution: y(x)=Θ(x)  f(x)  x , where Θ(x)=Θ(x + 1) is an arbitrary periodic function with period 1. For Θ(x) ≡ const, we have a particular solution y(x)=C  f(x)  x ,whereC is an arbitrary constant. 11. y(x + a) – by(x) =0. Solution: y(x)=Θ(x)b x/a , where Θ(x)=Θ(x + a) is an arbitrary periodic function with period a. For Θ(x) ≡ const, we have particular solution y(x)=Cb x/a ,whereC is an arbitrary constant. T12.1. LINEAR FUNCTIONAL EQUATIONS IN ONE INDEPENDENT VARIABLE 1413 12. y(x + a) – by(x) = f (x). 1 ◦ . Solution: y(x)=Θ(x)b x/a + ¯y(x), where Θ(x)=Θ(x + a) is an arbitrary periodic function with period a,and¯y(x)isany particular solution of the nonhomogeneous equation. 2 ◦ .Forf(x)= n  k=0 A k x n and b ≠ 1, the nonhomogeneous equation has a particular solution of the form ¯y(x)= n  k=0 B k x n ; the constants B k are found by the method of undetermined coefficients. 3 ◦ .Forf(x)= n  k=1 A k exp(λ k x), the nonhomogeneous equation has a particular solution of the form ¯y(x)= n  k=1 B k exp(λ k x); the constants B k are found by the method of undetermined coefficients. 4 ◦ .Forf(x)= n  k=1 A k cos(λ k x), the nonhomogeneous equation has a particular solution of the form ¯y(x)= n  k=1 B k cos(λ k x)+ n  k=1 D k sin(λ k x); the constants B k and D k are found by the method of undetermined coefficients. 5 ◦ .Forf(x)= n  k=1 A k sin(λ k x), the nonhomogeneous equation has a particular solution of the form ¯y(x)= n  k=1 B k cos(λ k x)+ n  k=1 D k sin(λ k x); the constants B k and D k are found by the method of undetermined coefficients. 13. y(x + a) – bxy(x) =0, a, b >0. Solution: y(x)=Θ(x)  ∞ 0 t (x/a)–1 e –t/(ab) dt, where Θ(x)=Θ(x + a) is an arbitrary periodic function with period a. 14. y(x + a) – f (x)y(x) =0. Here, f(x)=f (x + a) is a given periodic function with period a. Solution: y(x)=Θ(x)  f(x)  x/a , where Θ(x)=Θ(x + a) is an arbitrary periodic function with period a. For Θ(x) ≡ const, we have a particular solution y(x)=C  f(x)  x/a ,whereC is an arbitrary constant. 1414 FUNCTIONAL EQUATIONS T12.1.1-2. Linear functional equations involving y(x)andy(ax). 15. y(ax) – by(x) =0, a, b >0. Solution for x > 0: y(x)=x λ Θ  ln x ln a  , λ = ln b ln a , where Θ(z)=Θ(z + 1) is an arbitrary periodic function with period 1, a ≠ 1. For Θ(z) ≡ const, we have a particular solution y(x)=Cx λ ,whereC is an arbitrary constant. 16. y(ax) – by(x) = f(x). 1 ◦ . Solution: y(x)=Y (x)+¯y(x), where Y (x) is the general solution of the homogeneous equation Y (ax)–bY (x)=0 (see the previous equation), and ¯y(x) is any particular solution of the nonhomogeneous equation. 2 ◦ .Forf(x)= n  k=0 A k x n , the nonhomogeneous equation has a particular solution of the form ¯y(x)= n  k=0 A k a k – b x k , a k – b ≠ 0. 3 ◦ .Forf(x)=lnx n  k=0 A k x k , the nonhomogeneous equation has a particular solution of the form ¯y(x)= n  k=1 x k  B k ln x + C k  , B k = A k a k – b , C k =– A k a k ln a (a k – b) 2 . 17. y(2x) – a cos xy(x) =0. Solution for a > 0 and x > 0: y(x)=x ln a ln 2 – 1 sin x Θ  ln x ln 2  , where Θ(x)=Θ(x + 1) is an arbitrary periodic function with period 1. T12.1.1-3. Linear functional equations involving y(x)andy(a – x). 18. y(x) = y(–x). This functional equation may be treated as a definition of even functions. Solution: y(x)= ϕ(x)+ϕ(–x) 2 , where ϕ(x) is an arbitrary function. T12.1. LINEAR FUNCTIONAL EQUATIONS IN ONE INDEPENDENT VARIABLE 1415 19. y(x) =–y(–x). This functional equation may be treated as a definition of odd functions. Solution: y(x)= ϕ(x)–ϕ(–x) 2 , where ϕ(x) is an arbitrary function. 20. y(x) – y(a – x) =0. 1 ◦ . Solution: y(x)=Φ(x, a – x), where Φ(x, z)=Φ(z, x) is any symmetric function with two arguments. 2 ◦ . Specific particular solutions may be obtained using the formula y(x)=Ψ  ϕ(x)+ϕ(a – x)  by specifying the functions Ψ(z)andϕ(x). 21. y(x) + y(a – x) =0. 1 ◦ . Solution: y(x)=Φ(x, a – x), where Φ(x, z)=–Φ(z, x) is any antisymmetric function with two arguments. 2 ◦ . Specific particular solutions may be obtained using the formula y(x)=(2x – a)Ψ  ϕ(x)+ϕ(a – x)  by specifying Ψ(z)andϕ(x). 22. y(x) + y(a – x) = b. Solution: y(x)= 1 2 b + Φ(x, a – x), where Φ(x, z)=–Φ(z, x) is any antisymmetric function with two arguments. Particular solutions: y(x)=b sin 2  πx 2a  and y(x)=b cos 2  πx 2a  . 23. y(x) + y(a – x) = f(x). Here, the function f (x) must satisfy the condition f (x)=f(a – x). Solution: y(x)= 1 2 f(x)+Φ(x, a – x), where Φ(x, z)=–Φ(z, x) is any antisymmetric function with two arguments. 24. y(x) – y(a – x) = f(x). Here, the function f (x) must satisfy the condition f (x)=–f(a – x). Solution: y(x)= 1 2 f(x)+Φ(x, a – x), where Φ(x, z)=Φ(z, x) is any symmetric function with two arguments. 25. y(x) + g(x)y(a – x) = f (x). Solution: y(x)= f(x)–g(x)f (a – x) 1 – g(x)g(a – x) [if g(x)g(a – x) 1]. 1416 FUNCTIONAL EQUATIONS T12.1.1-4. Linear functional equations involving y(x)andy(a/x). 26. y(x) – y(a/x) =0. Babbage equation. Solution: y(x)=Φ(x, a/x), where Φ(x, z)=Φ(z, x) is any symmetric function with two arguments. 27. y(x) + y(a/x) =0. Solution: y(x)=Φ(x, a/x), where Φ(x, z)=–Φ(z, x) is any antisymmetric function with two arguments. 28. y(x) + y(a/x) = b. Solution: y(x)= 1 2 b + Φ(x, a/x), where Φ(x, z)=–Φ(z, x) is any antisymmetric function with two arguments. 29. y(x) + y(a/x) = f(x). The right-hand side must satisfy the condition f (x)=f (a/x). Solution: y(x)= 1 2 f(x)+Φ(x, a/x), where Φ(x, z)=–Φ(z, x) is any antisymmetric function with two arguments. 30. y(x) – y(a/x) = f(x). Here, the function f (x) must satisfy the condition f (x)=–f(a/x). Solution: y(x)= 1 2 f(x)+Φ(x, a/x), where Φ(x, z)=Φ(z, x) is any symmetric function with two arguments. 31. y(x) + x a y(1/x) =0. Solution: y(x)=(1 – x a )Φ(x, 1/x), where Φ(x, z)=Φ(z, x) is any symmetric function with two arguments. 32. y(x) – x a y(1/x) =0. Solution: y(x)=(1 + x a )Φ(x, 1/x), where Φ(x, z)=Φ(z, x) is any symmetric function with two arguments. 33. y(x) + g(x)y(a/x) = f(x). Solution: y(x)= f(x)–g(x)f(a/x) 1 – g(x)g(a/x) [if g(x)g(a/x) 1]. . with period a ,and y(x)isany particular solution of the nonhomogeneous equation. 2 ◦ .Forf(x)= n  k=0 A k x n and b ≠ 1, the nonhomogeneous equation has a particular solution of the form ¯y(x)= n  k=0 B k x n ;. equation has a particular solution of the form ¯y(x)= n  k=0 A k a k – b x k , a k – b ≠ 0. 3 ◦ .Forf(x)=lnx n  k=0 A k x k , the nonhomogeneous equation has a particular solution of the form ¯y(x)= n  k=1 x k  B k ln. the constants B k and D k are found by the method of undetermined coefficients. 5 ◦ .Forf(x)= n  k=1 A k sin(λ k x), the nonhomogeneous equation has a particular solution of the form ¯y(x)= n  k=1 B k cos(λ k x)+ n  k=1 D k sin(λ k x);