Page 428 - Handbook Of Integral Equations

b

6.8-3. Equations of the Form y(x)+ K(x, t)G t, y(t) dt = F (x)
a
b

27. y(x)+ f t, y(t) dt = g(x).
a
A solution: y(x)= g(x)+ λ, where λ is determined by the algebraic (or transcendental)
equation
b

λ + F(λ)=0, F(λ)= f t, g(t)+ λ dt.
a
b

28. y(x)+ e λ(x–t) f t, y(t) dt = g(x).
a
A solution: y(x)= βe λx + g(x), where λ is determined by the algebraic (or transcendental)
equation
b

β + F(β)=0, F(β)= e –λt f t, βe λt + g(t) dt.
a
b

29. y(x)+ g(x)f t, y(t) dt = h(x).
a
A solution: y(x)= λg(x)+ h(x), where λ is determined by the algebraic (or transcendental)
equation
b

λ + F(λ)=0, F(λ)= f t, λg(t)+ h(t) dt.
a
b

30. y(x)+ (Ax + Bt)f t, y(t) dt = g(x).
a
A solution: y(x)= g(x)+ λx + µ, where the constants λ and µ are determined from the
algebraic (or transcendental) system
b b

λ + A f t, g(t)+ λt + µ dt =0, µ + B tf t, g(t)+ λt + µ dt =0.
a a
b

31. y(x)+ cosh(λx + µt)f t, y(t) dt = h(x).
a
Using the formula cosh(λx + µt) = cosh(λx) cosh(µt) + sinh(µt) sinh(λx), we arrive at an
equation of the form 6.8.39:
b

y(x)+ cosh(λx)f 1 t, y(t) + sinh(λx)f 2 t, y(t) dt = h(x),
a

f 1 t, y(t) = cosh(µt)f t, y(t) , f 2 t, y(t) = sinh(µt)f t, y(t) .
b

32. y(x)+ sinh(λx + µt)f t, y(t) dt = h(x).
a
Using the formula sinh(λx + µt) = cosh(λx) sinh(µt) + cosh(µt) sinh(λx), we arrive at an
equation of the form 6.8.39:
b

y(x)+ cosh(λx)f 1 t, y(t) + sinh(λx)f 2 t, y(t) dt = h(x),
a

f 1 t, y(t) = sinh(µt)f t, y(t) , f 2 t, y(t) = cosh(µt)f t, y(t) .

© 1998 by CRC Press LLC

© 1998 by CRC Press LLC
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