Page 33 - Handbook Of Integral Equations

x
y(t) dt
40. √ = f(x).
2
a x – t 2
2 d x tf(t) dt
Solution: y = √ .
π dx 2 2
a x – t
•
Reference: P. P. Zabreyko, A. I. Koshelev, et al. (1975).
x y(t) dt
41. √ = f(x), a >0, a + b >0.
2
0 ax + bt 2
n
1 . For a polynomial right-hand side, f(x)= N A n x , the solution has the form
◦
n=0
N 1 n
A n n t dt
y(x)= x , B n = √ .
B n 0 a + bt 2
n=0

n
2 .For f(x)= x λ N A n x , where λ is an arbitrary number (λ > –1), the solution has the
◦
n=0
form
N 1 λ+n
A n n t dt
λ
y(x)= x x , B n = √ .
B n 0 a + bt 2
n=0

N
3 .For f(x)=ln x A n x n , the solution has the form
◦
n=0
N N 1 n 1 n
t dt t ln t
A n n A n C n n
y(x)=ln x x – 2 x , B n = √ , C n = √ dt.
B n B n 0 a + bt 2 0 a + bt 2
n=0 n=0
n
N
◦
4 .For f(x)= A n ln x) , the solution of the equation has the form
n=0
N

y(x)= A n Y n (x),
n=0
where the functions Y n = Y n (x) are given by
n λ 1 λ
d x z dz
Y n (x)= , I(λ)= √ .
dλ n I(λ) a + bz 2
λ=0 0
5 .For f(x)= N A n cos(λ n ln x)+ N B n sin(λ n ln x), the solution of the equation has the
◦
n=1 n=1
form
N N

y(x)= C n cos(λ n ln x)+ D n sin(λ n ln x),
n=1 n=1
where the constants C n and D n are found by the method of undetermined coefficients.

© 1998 by CRC Press LLC

© 1998 by CRC Press LLC
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