No solution to general Volterra integral equation of the second type?I encountered a (likely simple) Volterra-style integral equation as part of my work. For an arbitrary function p(s) which is square integrable, I have the following equation relating p(s) and its integral with respect to the real-to-real function Q:p(x)−C∫0xp(s)dQ(s)=0where C is a known real constant.I would hopefully like to show, without further assumptions, that this system has only the solution p(s)=0. I have very little training in differential equations, so I'm a bit lost as where to begin.

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No solution to general Volterra integral equation of the second type?I encountered a (likely simple) Volterra-style integral equation as part of my work. For an arbitrary function p(s) which is square integrable, I have the following equation relating p(s) and its integral with respect to the real-to-real function Q:p(x)−C∫0xp(s)dQ(s)=0where C is a known real constant.I would hopefully like to show, without further assumptions, that this system has only the solution p(s)=0. I have very little training in differential equations, so I&#039;m a bit lost as where to begin.

icebox2686zsd · Accepted Answer

Step 1Write the equation asp(x)=∫0xp(s)q(s)dswhere q(s)=CQ′(s). The equation shows that p(x) is differentiable, since it is equal to an antiderivative. Therefore by differentiatingp′(x)=p(x)q(x).and also p(0)=∫00p(s)q(s)ds=0.Step 2To prove from this that p(x)=0 for all x, one can use the following trick: Set r(x)=p(x)e−CQ(x). Thenr′(x)=p′(x)e−CQ(x)+p(x)e−CQ(x)⋅(−CQ′(x))=e−CQ(x)⋅(p′(x)−p(x)q(x))=0So r(x)=constan⁡t=r(0)=0 and therefore also q(x)=0 for all x.

No solution to general Volterra integral equation of the second type? I encountered a (likely si

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