# Integral Equations

solution Solve $u(x)=f(x)+\int _{0}^{x}k(x-y)u(y)\,dy$

solution Solve $u(x)=x+\int _{0}^{x}(x-y)u(y)\,dy$

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

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

solution Approximate $y(x)=x^{2}+\int _{0}^{1}\sin(xz)y(z)dz\,$

solution Find lambda: $y(x)=\lambda \int _{0}^{1}y(t)dt\,$ if $y(x)=c\,$

solution Find lambda: $y(x)=\lambda \int _{0}^{1}xty(t)dt\,$ if $y(x)=x\,$

solution Find lambda: $y(x)=\lambda \int _{0}^{1}(x^{2}-z^{2})y(z)dz\,$

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

solution Solve $y(x)=\lambda \int _{0}^{1}e^{{x+z}}y(z)dz,y(x)=e^{x}\,$

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

solution Solve $f(x)=\lambda \int _{0}^{2}\pi \sin(x+t)y(t)dt\,$

solution Solve $f(x)+\lambda \int _{0}^{1}xe^{z}f(z)dz\,$

solution Find the Euler equation $J(u)=\int \int _{{\mathbb {R}}}(y^{2}u_{x}^{2}+y^{2}uy^{2})dxdy\,$

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

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

solution Find the value of lambda for which the homogeneous Fredholm integral equation $y(x)=\lambda \int _{0}^{1}e^{x}e^{t}y(t)dt\,$ has a nontrivial solution, and find all the solutions.

solution Determine all values of the constants $a,b,c\,$ for which the integral equation $\int _{0}^{1}(1-xt)y(t)dt=ax^{2}+bx+c\,$ has solutions.

solution Solve: $g(s)=f(s)+\lambda \int _{0}^{{2\pi }}\sin(s)\cos(t)g(t)dt\,$

solution Solve: $u(x)=\int _{0}^{x}e^{{x-y}}u(y)dy\,$

solution Solve: $\sin(x)=\int _{0}^{x}e^{{x-t}}u(t)dt\,$

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

INTEGRAL EQUATIONS BOOKS