# Integral Equations

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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