2 ◦ .Forλ(2A + λ)=–k 2 < 0, the general solution of equation (1) is given by y(x)=C 1 cosh(kx)+C 2 sinh(kx)+f(x) – 2Aλ k  x a sinh[k(x – t)] f(t) dt, (3) where C 1 and C 2 are arbitrary constants. For λ(2A + λ)=k 2 > 0, the general solution of equation (1) is given by y(x)=C 1 cos(kx)+C 2 sin(kx)+f(x) – 2Aλ k  x a sin[k(x – t)] f(t) dt. (4) For λ =2A, the general solution of equation (1) is given by y(x)=C 1 + C 2 x + f(x)+4A 2  x a (x – t)f(t) dt. (5) The constants C 1 and C 2 in solutions (3)–(5) are determined by conditions (2). 30. y(x)+A  b a t sin(λ|x – t|)y(t) dt = f(x). This is a special case of equation 4.9.39 with g(t)=At. The solution of the integral equation can be written via the Bessel functions (or modified Bessel functions) of order 1/3. 31. y(x)+A  b a sin 3 (λ|x – t|)y(t) dt = f(x). Using the formula sin 3 β = – 1 4 sin 3β + 3 4 sin β, we arrive at an equation of the form 4.5.32 with n =2: y(x)+  b a  – 1 4 A sin(3λ|x – t|)+ 3 4 A sin(λ|x – t|)  y(t) dt = f(x). 32. y(x)+  b a  n  k=1 A k sin(λ k |x – t|)  y(t) dt = f (x), –∞ < a < b < ∞. 1 ◦ . Let us remove the modulus in the kth summand of the integrand: I k (x)=  b a sin(λ k |x – t|)y(t) dt =  x a sin[λ k (x – t)]y(t) dt +  b x sin[λ k (t – x)]y(t) dt. (1) Differentiating (1) with respect to x twice yields I  k = λ k  x a cos[λ k (x – t)]y(t) dt – λ k  b x cos[λ k (t – x)]y(t) dt, I  k =2λ k y(x) – λ 2 k  x a sin[λ k (x – t)]y(t) dt – λ 2 k  b x sin[λ k (t – x)]y(t) dt, (2) where the primes denote the derivatives with respect to x. By comparing formulas (1) and (2), we find the relation between I  k and I k : I  k =2λ k y(x) – λ 2 k I k , I k = I k (x). (3) Page 286 © 1998 by CRC Press LLC © 1998 by CRC Press LLC 2 ◦ . With the aid of (1), the integral equation can be rewritten in the form y(x)+ n  k=1 A k I k = f(x). (4) Differentiating (4) with respect to x twice and taking into account (3), we find that y  xx (x)+σ n y(x) – n  k=1 A k λ 2 k I k = f  xx (x), σ n =2 n  k=1 A k λ k . (5) Eliminating the integral I n from (4) and (5) yields y  xx (x)+(σ n + λ 2 n )y(x)+ n–1  k=1 A k (λ 2 n – λ 2 k )I k = f  xx (x)+λ 2 n f(x). (6) Differentiating (6) with respect to x twice and eliminating I n–1 from the resulting equation with the aid of (6), we obtain a similar equation whose left-hand side is a second-order linear differential operator (acting on y) with constant coefficients plus the sum n–2  k=1 B k I k .Ifwe successively eliminate I n–2 , I n–3 , , with the aid of double differentiation, then we finally arrive at a linear nonhomogeneous ordinary differential equation of order 2n with constant coefficients. 3 ◦ . The boundary conditions for y(x) can be found by setting x = a in the integral equation and all its derivatives. (Alternatively, these conditions can be found by setting x = a and x = b in the integral equation and all its derivatives obtained by means of double differentiation.) 33. y(x) – λ  ∞ –∞ sin(x – t) x – t y(t) dt = f (x). Solution: y(x)=f(x)+ λ √ 2π – πλ  ∞ –∞ sin(x – t) x – t f(t) dt, λ ≠  2 π . • Reference: F. D. Gakhov and Yu. I. Cherskii (1978). 4.5-3. Kernels Containing Tangent 34. y(x) – λ  b a tan(βx)y(t) dt = f (x). This is a special case of equation 4.9.1 with g(x) = tan(βx) and h(t)=1. 35. y(x) – λ  b a tan(βt)y(t) dt = f (x). This is a special case of equation 4.9.1 with g(x) = 1 and h(t) = tan(βt). 36. y(x) – λ  b a [A tan(βx)+B tan(βt)]y(t) dt = f (x). This is a special case of equation 4.9.4 with g(x) = tan(βx). Page 287 © 1998 by CRC Press LLC © 1998 by CRC Press LLC 37. y(x) – λ  b a tan(βx) tan(βt) y(t) dt = f (x). This is a special case of equation 4.9.1 with g(x) = tan(βx) and h(t)= 1 tan(βt) . 38. y(x) – λ  b a tan(βt) tan(βx) y(t) dt = f (x). This is a special case of equation 4.9.1 with g(x)= 1 tan(βx) and h(t) = tan(βt). 39. y(x) – λ  b a tan k (βx) tan m (µt)y(t) dt = f (x). This is a special case of equation 4.9.1 with g(x) = tan k (βx) and h(t) = tan m (µt). 40. y(x) – λ  b a t k tan m (βx)y(t) dt = f (x). This is a special case of equation 4.9.1 with g(x) = tan m (βx) and h(t)=t k . 41. y(x) – λ  b a x k tan m (βt)y(t) dt = f (x). This is a special case of equation 4.9.1 with g(x)=x k and h(t) = tan m (βt). 42. y(x) – λ  b a [A + B(x – t) tan(βt)]y(t) dt = f (x). This is a special case of equation 4.9.8 with h(t) = tan(βt). 43. y(x) – λ  b a [A + B(x – t) tan(βx)]y(t) dt = f (x). This is a special case of equation 4.9.10 with h(x) = tan(βx). 4.5-4. Kernels Containing Cotangent 44. y(x) – λ  b a cot(βx)y(t) dt = f (x). This is a special case of equation 4.9.1 with g(x) = cot(βx) and h(t)=1. 45. y(x) – λ  b a cot(βt)y(t) dt = f (x). This is a special case of equation 4.9.1 with g(x) = 1 and h(t) = cot(βt). 46. y(x) – λ  b a [A cot(βx)+B cot(βt)]y(t) dt = f (x). This is a special case of equation 4.9.4 with g(x) = cot(βx). Page 288 © 1998 by CRC Press LLC © 1998 by CRC Press LLC 47. y(x) – λ  b a cot(βx) cot(βt) y(t) dt = f (x). This is a special case of equation 4.9.1 with g(x) = cot(βx) and h(t)= 1 cot(βt) . 48. y(x) – λ  b a cot(βt) cot(βx) y(t) dt = f (x). This is a special case of equation 4.9.1 with g(x)= 1 cot(βx) and h(t) = cot(βt). 49. y(x) – λ  b a cot k (βx) cot m (µt)y(t) dt = f (x). This is a special case of equation 4.9.1 with g(x) = cot k (βx) and h(t) = cot m (µt). 50. y(x) – λ  b a t k cot m (βx)y(t) dt = f (x). This is a special case of equation 4.9.1 with g(x) = cot m (βx) and h(t)=t k . 51. y(x) – λ  b a x k cot m (βt)y(t) dt = f (x). This is a special case of equation 4.9.1 with g(x)=x k and h(t) = cot m (βt). 52. y(x) – λ  b a [A + B(x – t) cot(βt)]y(t) dt = f (x). This is a special case of equation 4.9.8 with h(t) = cot(βt). 53. y(x) – λ  b a [A + B(x – t) cot(βx)]y(t) dt = f (x). This is a special case of equation 4.9.10 with h(x) = cot(βx). 4.5-5. Kernels Containing Combinations of Trigonometric Functions 54. y(x) – λ  b a cos k (βx) sin m (µt)y(t) dt = f (x). This is a special case of equation 4.9.1 with g(x) = cos k (βx) and h(t) = sin m (µt). 55. y(x) – λ  b a [A sin(αx) cos(βt)+B sin(γx) cos(δt)]y(t) dt = f (x). This is a special case of equation 4.9.18 with g 1 (x) = sin(αx), h 1 (t)=A cos(βt), g 2 (x) = sin(γx), and h 2 (t)=B cos(δt). 56. y(x) – λ  b a tan k (γx) cot m (µt)y(t) dt = f (x). This is a special case of equation 4.9.1 with g(x) = tan k (γx) and h(t) = cot m (µt). 57. y(x) – λ  b a [A tan(αx) cot(βt)+B tan(γx) cot(δt)]y(t) dt = f (x). This is a special case of equation 4.9.18 with g 1 (x) = tan(αx), h 1 (t)=A cot(βt), g 2 (x) = tan(γx), and h 2 (t)=B cot(δt). Page 289 © 1998 by CRC Press LLC © 1998 by CRC Press LLC 4.5-6. A Singular Equation 58. Ay(x) – B 2π  2π 0 cot  t – x 2  y(t) dt = f (x), 0 ≤ x ≤ 2π. Here the integral is understood in the sense of the Cauchy principal value. Without loss of generality we may assume that A 2 + B 2 =1. Solution: y(x)=Af(x)+ B 2π  2π 0 cot  t – x 2  f(t) dt + B 2 2πA  2π 0 f(t) dt. • Reference: I. K. Lifanov (1996). 4.6. Equations Whose Kernels Contain Inverse Trigonometric Functions 4.6-1. Kernels Containing Arccosine 1. y(x) – λ  b a arccos(βx)y(t) dt = f (x). This is a special case of equation 4.9.1 with g(x) = arccos(βx) and h(t)=1. 2. y(x) – λ  b a arccos(βt)y(t) dt = f (x). This is a special case of equation 4.9.1 with g(x) = 1 and h(t) = arccos(βt). 3. y(x) – λ  b a arccos(βx) arccos(βt) y(t) dt = f (x). This is a special case of equation 4.9.1 with g(x) = arccos(βx) and h(t)= 1 arccos(βt) . 4. y(x) – λ  b a arccos(βt) arccos(βx) y(t) dt = f (x). This is a special case of equation 4.9.1 with g(x)= 1 arccos(βx) and h(t) = arccos(βt). 5. y(x) – λ  b a arccos k (βx) arccos m (µt)y(t) dt = f (x). This is a special case of equation 4.9.1 with g(x) = arccos k (βx) and h(t) = arccos m (µt). 6. y(x) – λ  b a t k arccos m (βx)y(t) dt = f (x). This is a special case of equation 4.9.1 with g(x) = arccos m (βx) and h(t)=t k . 7. y(x) – λ  b a x k arccos m (βt)y(t) dt = f (x). This is a special case of equation 4.9.1 with g(x)=x k and h(t) = arccos m (βt). Page 290 © 1998 by CRC Press LLC © 1998 by CRC Press LLC 8. y(x) – λ  b a [A + B(x – t) arccos(βx)]y(t) dt = f(x). This is a special case of equation 4.9.10 with h(x) = arccos(βx). 9. y(x) – λ  b a [A + B(x – t) arccos(βt)]y(t) dt = f(x). This is a special case of equation 4.9.8 with h(t) = arccos(βt). 4.6-2. Kernels Containing Arcsine 10. y(x) – λ  b a arcsin(βx)y(t) dt = f (x). This is a special case of equation 4.9.1 with g(x) = arcsin(βx) and h(t)=1. 11. y(x) – λ  b a arcsin(βt)y(t) dt = f (x). This is a special case of equation 4.9.1 with g(x) = 1 and h(t) = arcsin(βt). 12. y(x) – λ  b a arcsin(βx) arcsin(βt) y(t) dt = f (x). This is a special case of equation 4.9.1 with g(x) = arcsin(βx) and h(t)= 1 arcsin(βt) . 13. y(x) – λ  b a arcsin(βt) arcsin(βx) y(t) dt = f (x). This is a special case of equation 4.9.1 with g(x)= 1 arcsin(βx) and h(t) = arcsin (βt). 14. y(x) – λ  b a arcsin k (βx) arcsin m (µt)y(t) dt = f (x). This is a special case of equation 4.9.1 with g(x) = arcsin k (βx) and h(t) = arcsin m (µt). 15. y(x) – λ  b a t k arcsin m (βx)y(t) dt = f (x). This is a special case of equation 4.9.1 with g(x) = arcsin m (βx) and h(t)=t k . 16. y(x) – λ  b a x k arcsin m (βt)y(t) dt = f (x). This is a special case of equation 4.9.1 with g(x)=x k and h(t) = arcsin m (βt). 17. y(x) – λ  b a [A + B(x – t) arcsin(βt)]y(t) dt = f(x). This is a special case of equation 4.9.8 with h(t) = arcsin(βt). 18. y(x) – λ  b a [A + B(x – t) arcsin(βx)]y(t) dt = f(x). This is a special case of equation 4.9.10 with h(x) = arcsin(βx). Page 291 © 1998 by CRC Press LLC © 1998 by CRC Press LLC 4.6-3. Kernels Containing Arctangent 19. y(x) – λ  b a arctan(βx)y(t) dt = f (x). This is a special case of equation 4.9.1 with g(x) = arctan(βx) and h(t)=1. 20. y(x) – λ  b a arctan(βt)y(t) dt = f (x). This is a special case of equation 4.9.1 with g(x) = 1 and h(t) = arctan(βt). 21. y(x) – λ  b a [A arctan(βx)+B arctan(βt)]y(t) dt = f (x). This is a special case of equation 4.9.4 with g(x) = arctan(βx). 22. y(x) – λ  b a arctan(βx) arctan(βt) y(t) dt = f (x). This is a special case of equation 4.9.1 with g(x) = arctan(βx) and h(t)= 1 arctan(βt) . 23. y(x) – λ  b a arctan(βt) arctan(βx) y(t) dt = f (x). This is a special case of equation 4.9.1 with g(x)= 1 arctan(βx) and h(t) = arctan(βt). 24. y(x) – λ  b a arctan k (βx) arctan m (µt)y(t) dt = f (x). This is a special case of equation 4.9.1 with g(x) = arctan k (βx) and h(t) = arctan m (µt). 25. y(x) – λ  b a t k arctan m (βx)y(t) dt = f (x). This is a special case of equation 4.9.1 with g(x) = arctan m (βx) and h(t)=t k . 26. y(x) – λ  b a x k arctan m (βt)y(t) dt = f (x). This is a special case of equation 4.9.1 with g(x)=x k and h(t) = arctan m (βt). 27. y(x) – λ  b a [A + B(x – t) arctan(βt)]y(t) dt = f(x). This is a special case of equation 4.9.8 with h(t) = arctan(βt). 28. y(x) – λ  b a [A + B(x – t) arctan(βx)]y(t) dt = f(x). This is a special case of equation 4.9.10 with h(x) = arctan(βx). Page 292 © 1998 by CRC Press LLC © 1998 by CRC Press LLC 4.6-4. Kernels Containing Arccotangent 29. y(x) – λ  b a arccot(βx)y(t) dt = f (x). This is a special case of equation 4.9.1 with g(x) = arccot(βx) and h(t)=1. 30. y(x) – λ  b a arccot(βt)y(t) dt = f (x). This is a special case of equation 4.9.1 with g(x) = 1 and h(t) = arccot(βt). 31. y(x) – λ  b a [A arccot(βx)+B arccot(βt)]y(t) dt = f (x). This is a special case of equation 4.9.4 with g(x) = arccot(βx). 32. y(x) – λ  b a arccot(βx) arccot(βt) y(t) dt = f (x). This is a special case of equation 4.9.1 with g(x) = arccot(βx) and h(t)= 1 arccot(βt) . 33. y(x) – λ  b a arccot(βt) arccot(βx) y(t) dt = f (x). This is a special case of equation 4.9.1 with g(x)= 1 arccot(βx) and h(t) = arccot(βt). 34. y(x) – λ  b a arccot k (βx) arccot m (µt)y(t) dt = f (x). This is a special case of equation 4.9.1 with g(x) = arccot k (βx) and h(t) = arccot m (µt). 35. y(x) – λ  b a t k arccot m (βx)y(t) dt = f (x). This is a special case of equation 4.9.1 with g(x) = arccot m (βx) and h(t)=t k . 36. y(x) – λ  b a x k arccot m (βt)y(t) dt = f (x). This is a special case of equation 4.9.1 with g(x)=x k and h(t) = arccot m (βt). 37. y(x) – λ  b a [A + B(x – t) arccot(βt)]y(t) dt = f(x). This is a special case of equation 4.9.8 with h(t) = arccot(βt). 38. y(x) – λ  b a [A + B(x – t) arccot(βx)]y(t) dt = f(x). This is a special case of equation 4.9.10 with h(x) = arccot(βx). Page 293 © 1998 by CRC Press LLC © 1998 by CRC Press LLC 4.7. Equations Whose Kernels Contain Combinations of Elementary Functions 4.7-1. Kernels Containing Exponential and Hyperbolic Functions 1. y(x) – λ  b a e µ(x–t) cosh[β(x – t)]y(t) dt = f (x). This is a special case of equation 4.9.18 with g 1 (x)=e µx cosh(βx), h 1 (t)=e –µt cosh(βt), g 2 (x)=e µx sinh(βx), and h 2 (t)=–e –µt sinh(βt). 2. y(x) – λ  b a e µ(x–t) sinh[β(x – t)]y(t) dt = f (x). This is a special case of equation 4.9.18 with g 1 (x)=e µx sinh(βx), h 1 (t)=e –µt cosh(βt), g 2 (x)=e µx cosh(βx), and h 2 (t)=–e –µt sinh(βt). 3. y(x) – λ  b a te µ(x–t) sinh[β(x – t)]y(t) dt = f (x). This is a special case of equation 4.9.18 with g 1 (x)=e µx sinh(βx), h 1 (t)=te –µt cosh(βt), g 2 (x)=e µx cosh(βx), and h 2 (t)=–te –µt sinh(βt). 4.7-2. Kernels Containing Exponential and Logarithmic Functions 4. y(x) – λ  b a e µt ln(βx)y(t) dt = f (x). This is a special case of equation 4.9.1 with g(x) = ln(βx) and h(t)=e µt . 5. y(x) – λ  b a e µx ln(βt)y(t) dt = f (x). This is a special case of equation 4.9.1 with g(x)=e µx and h(t) = ln(βt). 6. y(x) – λ  b a e µ(x–t) ln(βx)y(t) dt = f (x). This is a special case of equation 4.9.1 with g(x)=e µx ln(βx) and h(t)=e –µt . 7. y(x) – λ  b a e µ(x–t) ln(βt)y(t) dt = f (x). This is a special case of equation 4.9.1 with g(x)=e µx and h(t)=e –µt ln(βt). 8. y(x) – λ  b a e µ(x–t) (ln x – ln t)y(t) dt = f (x). This is a special case of equation 4.9.18 with g 1 (x)=e µx ln x, h 1 (t)=e –µt , g 2 (x)=e µx , and h 2 (t)=–e –µt ln t. Page 294 © 1998 by CRC Press LLC © 1998 by CRC Press LLC 9. y(x)+ b 2 – a 2 2a  ∞ 0 1 t exp  –a    ln x t     y(t) dt = f (x). Solution with a >0,b > 0, and x >0: y(x)=f(x)+ a 2 – b 2 2b  ∞ 0 1 t exp  –b    ln x t     f(t) dt. • Reference: F. D. Gakhov and Yu. I. Cherskii (1978). 4.7-3. Kernels Containing Exponential and Trigonometric Functions 10. y(x) – λ  b a e µt cos(βx)y(t) dt = f (x). This is a special case of equation 4.9.1 with g(x) = cos(βx) and h(t)=e µt . 11. y(x) – λ  b a e µx cos(βt)y(t) dt = f (x). This is a special case of equation 4.9.1 with g(x)=e µx and h(t) = cos(βt). 12. y(x) – λ  ∞ 0 e µ(x–t) cos(xt)y(t) dt = f (x). Solution: y(x)= f(x) 1 – π 2 λ 2 + λ 1 – π 2 λ 2  ∞ 0 e µ(x–t) cos(xt)f(t) dt, λ ≠ ±  2/π. 13. y(x) – λ  b a e µ(x–t) cos[β(x – t)]y(t) dt = f (x). This is a special case of equation 4.9.18 with g 1 (x)=e µx cos(βx), h 1 (t)=e –µt cos(βt), g 2 (x)=e µx sin(βx), and h 2 (t)=e –µt sin(βt). 14. y(x) – λ  b a e µt sin(βx)y(t) dt = f (x). This is a special case of equation 4.9.1 with g(x) = sin(βx) and h(t)=e µt . 15. y(x) – λ  b a e µx sin(βt)y(t) dt = f (x). This is a special case of equation 4.9.1 with g(x)=e µx and h(t) = sin(βt). 16. y(x) – λ  ∞ 0 e µ(x–t) sin(xt)y(t) dt = f (x). Solution: y(x)= f(x) 1 – π 2 λ 2 + λ 1 – π 2 λ 2  ∞ 0 e µ(x–t) sin(xt)f(t) dt, λ ≠ ±  2/π. Page 295 © 1998 by CRC Press LLC © 1998 by CRC Press LLC [...]... b 50 tank (βx) lnm (µt)y(t) dt = f (x) y(x) – λ a This is a special case of equation 4.9.1 with g(x) = tank (βx) and h(t) = lnm (µt) b 51 tank (βt) lnm (µx)y(t) dt = f (x) y(x) – λ a This is a special case of equation 4.9.1 with g(x) = lnm (µx) and h(t) = tank (βt) b 52 cotk (βx) lnm (µt)y(t) dt = f (x) y(x) – λ a This is a special case of equation 4.9.1 with g(x) = cotk (βx) and h(t) = lnm (µt) b 53 ... (x) y(x) – λ a This is a special case of equation 4.9.1 with g(x) = lnm (µx) and h(t) = cothk (βt) 4.7 -5 Kernels Containing Hyperbolic and Trigonometric Functions b 34 coshk (βx) cosm (µt)y(t) dt = f (x) y(x) – λ a This is a special case of equation 4.9.1 with g(x) = coshk (βx) and h(t) = cosm (µt) b 35 coshk (βt) cosm (µx)y(t) dt = f (x) y(x) – λ a This is a special case of equation 4.9.1 with g(x) =... Y (z) + λ τ Jν (zτ )Y (τ ) dτ = F (z) 0 b 5 y(x) – λ [A + B(x – t)Jν (βt)]y(t) dt = f (x) a This is a special case of equation 4.9.8 with h(t) = Jν (βt) b 6 y(x) – λ [A + B(x – t)Jν (βx)]y(t) dt = f (x) a This is a special case of equation 4.9.10 with h(x) = Jν (βx) b 7 y(x) – λ [AJµ (αx) + BJν (βt)]y(t) dt = f (x) a This is a special case of equation 4.9 .5 with g(x) = AJµ (αx) and h(t) = BJν (βt)... h(x) = Yν (βx) b 13 y(x) – λ [AYµ (αx) + BYν (βt)]y(t) dt = f (x) a This is a special case of equation 4.9 .5 with g(x) = AYµ (αx) and h(t) = BYν (βt) b 14 y(x) – λ [AYµ (x)Yµ (t) + BYν (x)Yν (t)]y(t) dt = f (x) a This is a special case of equation 4.9.14 with g(x) = Yµ (x) and h(t) = Yν (t) b 15 y(x) – λ [AYµ (x)Yν (t) + BYν (x)Yµ (t)]y(t) dt = f (x) a This is a special case of equation 4.9.17 with... = Kν (βt) b 25 y(x) – λ [A + B(x – t)Kν (βt)]y(t) dt = f (x) a This is a special case of equation 4.9.8 with h(t) = Kν (βt) b 26 y(x) – λ [A + B(x – t)Kν (βx)]y(t) dt = f (x) a This is a special case of equation 4.9.10 with h(x) = Kν (βx) © 1998 by CRC Press LLC © 1998 by CRC Press LLC Page 301 b 27 y(x) – λ [AKµ (αx) + BKν (βt)]y(t) dt = f (x) a This is a special case of equation 4.9 .5 with g(x) =... Kµ (x) and h(t) = Kν (t) b 29 y(x) – λ [AKµ (x)Kν (t) + BKν (x)Kµ (t)]y(t) dt = f (x) a This is a special case of equation 4.9.17 with g(x) = Kµ (x) and h(t) = Kν (t) 4.9 Equations Whose Kernels Contain Arbitrary Functions 4.9-1 Equations With Degenerate Kernel: K(x, t) = g1 (x)h1 (t) + · · · + gn (x)hn (t) b 1 y(x) – λ g(x)h(t)y(t) dt = f (x) a 1◦ Assume that λ ≠ Solution: –1 b a g(t)h(t) dt –1... the equation has a unique solution, which is given by ∞ y(x) = f (x) + R(x – t)f (t) dt, –∞ 1 R(x) = √ 2π ∞ R(u)eiux du, R(u) = –∞ K(u) √ 1 – 2π K(u) • Reference: V A Ditkin and A P Prudnikov (19 65) ∞ 25 y(x) – K(x – t)y(t) dt = f (x) 0 The Wiener–Hopf equation of the second kind.* Here 0 ≤ x < ∞, K(x) ∈ L1 (–∞, ∞), f (x) ∈ L1 (0, ∞), and y(x) ∈ L1 (0, ∞) For the integral equation to be solvable,... LLC Page 319 4.9-3 Other Equations of the Form y(x) + b a K(x, t)y(t) dt = F (x) ∞ 26 y(x) – K(x + t)y(t) dt = f (x) –∞ The Fourier transform is used to solving this equation Solution: √ ∞ 1 f (u) + 2π f (–u)K(u) iux y(x) = √ e du, √ 2π –∞ 1 – 2π K(u)K(–u) where 1 f (u) = √ 2π ∞ ∞ 1 K(u) = √ 2π f (x)e–iux dx, –∞ K(x)e–iux dx –∞ • Reference: V A Ditkin and A P Prudnikov (19 65) ∞ 27 y(x) + eβt K(x +... case of equation 4.9.1 with g(x) = cotk (βx) and h(t) = lnm (µt) b 53 cotk (βt) lnm (µx)y(t) dt = f (x) y(x) – λ a This is a special case of equation 4.9.1 with g(x) = lnm (µx) and h(t) = cotk (βt) 4.8 Equations Whose Kernels Contain Special Functions 4.8-1 Kernels Containing Bessel Functions b 1 y(x) – λ Jν (βx)y(t) dt = f (x) a This is a special case of equation 4.9.1 with g(x) = Jν (βx) and h(t) =... of equation 4.9.1 with g(x) = cosm (µx) and h(t) = tanhk (βt) b 44 tanhk (βx) sinm (µt)y(t) dt = f (x) y(x) – λ a This is a special case of equation 4.9.1 with g(x) = tanhk (βx) and h(t) = sinm (µt) b 45 tanhk (βt) sinm (µx)y(t) dt = f (x) y(x) – λ a This is a special case of equation 4.9.1 with g(x) = sinm (µx) and h(t) = tanhk (βt) 4.7-6 Kernels Containing Logarithmic and Trigonometric Functions b . (x). This is a special case of equation 4.9.10 with h(x) = cot(βx). 4 .5- 5. Kernels Containing Combinations of Trigonometric Functions 54 . y(x) – λ  b a cos k (βx) sin m (µt)y(t) dt = f (x). This is. that y  xx (x)+σ n y(x) – n  k=1 A k λ 2 k I k = f  xx (x), σ n =2 n  k=1 A k λ k . (5) Eliminating the integral I n from (4) and (5) yields y  xx (x)+(σ n + λ 2 n )y(x)+ n–1  k=1 A k (λ 2 n – λ 2 k )I k =. I. Cherskii (1978). 4 .5- 3. Kernels Containing Tangent 34. y(x) – λ  b a tan(βx)y(t) dt = f (x). This is a special case of equation 4.9.1 with g(x) = tan(βx) and h(t)=1. 35. y(x) – λ  b a tan(βt)y(t)