# Volterra integral equation with variable boundaries

$$\phi (x)=x+\lambda \int_{a}^{x}(x-y)\phi (y)dy$$

I'm also Trying to solve this integral equation like she does https://math.stackexchange.com/questions/404959/solving-an-integral-equation-with-a-separable-kernel/408757#408757 and I also have some doubts about it. What to do? It's the first time I'm solving something like this, I only learned the matrix and determinant method for usual equation integrals. Can you explain how to solve this equation integrals? I searched the web but didn't get much anything to help me solve this. Thank a lot

• The techniques for solving integral equations vary greatly based on the form they take. What is the one you are trying to solve? Commented Jun 1, 2013 at 20:48
• Try differentiating w.r.t. $x$. Commented Jun 1, 2013 at 21:01
• Yes I did like in the other link.but what I do next? Commented Jun 1, 2013 at 21:29
• @Amccds: Are you taking a course in integral equations? Commented Jun 3, 2013 at 5:17

Related problem: (I), (II). You can use the method of successive approximation which can be summarized as

$$\phi_{n} (x)=x+\lambda \int_{a}^{x}(x-y)\phi_{n-1} (y)dy,$$

where $\phi_0(x)$ can be any real valued function. For simplicity, you can choose $\phi_0(x)=0$, then start to find $\phi_1,\phi_2,\dots$. For instance,

$$\phi_1(x)=x+\lambda \int_{a}^{x}(x-y)(0)dy = x.$$

$$\phi_{2} (x)=x+\lambda \int_{a}^{x}(x-y)\phi_{1} (y)dy$$

$$\phi_{2} (x)=x+\lambda \int_{a}^{x}(x-y)(y)dy = \dots.$$

If you can get a general formula for $\phi_n(x)$, then the solution $\phi(x)$ will be

$$\phi(x) = \lim_{n\to \infty} \phi_n(x).$$

Off course this process will impose some conditions on $\lambda$.

Another approach: Here is another approach. Differentiating the integral equation, we get

$$\phi (x)=x+\lambda \int_{a}^{x}(x-y)\phi (y)dy \implies \phi'(x)= 1+\lambda\int_{a}^{x}\phi(y) dy.$$

Differentiating again, we get the differential equation

$$\phi''(x) = \lambda \phi(x).$$

Solving the ode subject to the initial conditions $\phi(a)=a$ and $\phi'(a)=1$ gives the desired solution

$$\phi \left( x \right) = \frac{1}{2}\,{\frac { \left( \sqrt {\lambda}a+1 \right) {{\rm e}^{\sqrt {\lambda}x}}}{{{\rm e}^{\sqrt {\lambda}a}} \sqrt {\lambda}}}+\frac{1}{2}\,{\frac { \left( \sqrt {\lambda}a-1 \right) { {\rm e}^{-\sqrt {\lambda}x}}}{{{\rm e}^{-\sqrt {\lambda}a}}\sqrt { \lambda}}}.$$