Section 13.1

Examples of Problems with Solutions

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Problem 13.1

Construct the resolvent of the integral equation

x(t)=4∫0tet−sx(s)ds+e3t,t∈[0,T] x(t) = 4 \int _{0}^{t} e^{t-s} x(s) d s+e^{3 t}, t \in [0, T]

and apply it to solve the equation.

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Problem 13.2

Solve in the space L2([0,1])L_{2}(\left[0,1\right]) the integral equation with a degenerate kernel

x(t)=λ∫01(28t5s2−15t3s)x(s)ds+1t3,λ∈C x(t) = \lambda \int _{0}^{1}\left(28 t^{5} s^{2}-15 t^{3} s\right) x(s) d s+\frac{1}{\sqrt[3]{t}}, \lambda \in \mathbb {C}
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Problem 13.3

Find all values of the parameters p,q,rp, q, r, for which the integral equation x(t)=λ∫−11(1+2t+3st)x(s)ds+pt2+qt+r,t∈[−1,1]x(t) = \lambda \int_{-1}^{1}(1+2 t+3 s t) x(s) d s+p t^{2}+q t+r, t \in [-1,1], has a solution in the space L2([−1,1])L_{2}(\left[-1,1\right]) for every λ∈C\lambda \in \mathbb {C}.

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Problem 13.4

Find the characteristic numbers and corresponding normalized eigenfunctions and solve the integral equation with a symmetric kernel

x(t)=λ∫02π(cos⁡(t−s)+2sin⁡2tsin⁡2s)x(s)ds+cos⁡t,t∈[0,2π] x(t) = \lambda \int _{0}^{2 \pi }(\cos (t-s)+2 \sin 2 t \sin 2 s) x(s) d s+\cos t, t \in [0,2 \pi ]
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