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\,


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