# Integral Equations

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solution Solve $\displaystyle u(x) = f(x) + \int_0^x k(x-y)u(y)\,dy$

solution Solve $\displaystyle u(x) = x + \int_0^x (x-y)u(y)\,dy$

solution Formulate an integral equation from the BVP: $\displaystyle u'' + p(x)u' + q(x)u=f(x),\,\,\,x>a$

solution Solve $\displaystyle y''(x)=f(x), y(0)=A, y'(0)=B\,$

solution Approximate $\displaystyle y(x) = x^2 + \int_0^1 \sin(xz) y(z) dz\,$

solution Find lambda: $\displaystyle y(x)=\lambda\int_0^1 y(t)dt\,$ if $\displaystyle y(x)=c\,$

solution Find lambda: $\displaystyle y(x)=\lambda\int_0^1 x t y(t) dt\,$ if $\displaystyle y(x)=x\,$

solution Find lambda: $\displaystyle y(x)=\lambda \int_0^1 (x^2-z^2) y(z) dz\,$

solution Write as an ODE: $\displaystyle \int_a^b f(x,y) \sqrt{1+y'^2}\,dx\,$

solution Solve $\displaystyle y(x)=\lambda\int_0^1 e^{x+z} y(z) dz, y(x)=e^x\,$

solution Formulate an integral equation from the IVP: $\displaystyle y''-\lambda y=f(x), x>0, y(0)=1, y'(0)=0\,$

solution Solve $\displaystyle f(x) = \lambda \int_0^2\pi \sin(x+t) y(t)dt\,$

solution Solve $\displaystyle f(x) + \lambda \int_0^1 x e^z f(z) dz\,$

solution Find the Euler equation $\displaystyle J(u) = \int\int_\mathbb{R} (y^2 u_x^2 + y^2 u y^2) dx dy\,$

solution Reduce to a PDE: $\displaystyle J(u) = \int\int_\mathbb{R} dxdy\,$

solution Transform the BVP to an integral equation: $\displaystyle \frac{d^2y}{dx^2}+y=x, y(0)=0, y'(1)=0\,$ .

solution Find the value of lambda for which the homogeneous Fredholm integral equation $\displaystyle y(x) = \lambda\int_0^1 e^xe^ty(t)dt\,$ has a nontrivial solution, and find all the solutions.

solution Determine all values of the constants $\displaystyle a, b, c\,$ for which the integral equation $\displaystyle \int_0^1(1-xt)y(t)dt = ax^2+bx+c\,$ has solutions.

solution Solve: $\displaystyle g(s) = f(s) + \lambda \int_0^{2\pi} \sin(s) \cos(t) g(t) dt \,$

solution Solve: $\displaystyle u(x) = \int_0^x e^{x-y} u(y) dy\,$

solution Solve: $\displaystyle \sin(x) = \int_0^x e^{x-t}u(t)dt\,$

solution Convert to an integral equation: $\displaystyle y'' + y = \cos(x), y(0)=0, y'(0)=0\,$

INTEGRAL EQUATIONS BOOKS