Exact Solutions > Integral Equations > Volterra Integral Equations of the Second Kind and Related Linear Integral Equations with Variable Limit of Integration

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2. Volterra Integral Equations of the Second Kind

2-1. Integral equations with kernels involving power-law functions

1. y(x) − λ y(t) dt = f(x).
2. y(x) + λ (x − t) y(t) dt = f(x).
3. y(x) + λ (x − t)² y(t) dt = f(x).
4. y(x) + λ (x − t)³ y(t) dt = f(x).
5. y(x) + A (x − t)ⁿ y(t) dt = f(x).
6. y(x) + λ (x − t)^−1/2 y(t) dt = f(x).    Abel equation of the second kind.
7. y(x) − λ (x − t)^−α y(t) dt = f(x).    Generalized Abel equation of the second kind.

2-2. Integral equations with kernels involving exponential functions

8. y(x) + A e^{λ(x − t)} y(t) dt = f(x).
9. y(x) + A [e^{λ(x − t)} − 1] y(t) dt = f(x).
10. y(x) + A (x − t)e^{λ(x − t)} y(t) dt = f(x).

2-3. Integral equations with kernels involving hyperbolic or spacial functions

11. y(x) + A cosh[λ(x − t)] y(t) dt = f(x).
12. y(x) + A sinh[λ(x − t)] y(t) dt = f(x).
13. y(x) − λ J₀(x − t) y(t) dt = f(x).

2-4. Integral equations with kernels involving arbitrary functions

14. y(x) − g(x) h(t) y(t) dt = f(x).
15. y(x) + (x − t) g(x) y(t) dt = f(x).
16. y(x) + (x − t) g(t) y(t) dt = f(x).
17. y(x) + K(x − t) y(t) dt = f(x).    Renewal equation.