Help integrating an interpolating function inside NDSolve

Question

I am trying to solve an ODE having a term of integration of an interpolating function multiplied by the variable, as shown below. However, I get the following error:

Integrate::ilim: Invalid integration variable or limit(s) in {0.,0,4}.

Code:

data = Table[{x, x + Sin[2 Pi x]}, {x, 0, 4.82, 0.1}];
f = Interpolation[data];

NDSolve[{Integrate[f[x]*ϕ[x], {x, 0, 4}] + ϕ'[x] == 1, ϕ[0] == 0}, ϕ, {x, 0, 4}]

Thanks for your help in anticipation.

Answer 1

The issue is with the integral part Integrate[f[x]*ϕ[x], {x, 0, 4}]. To solve it as a stand alone definite integral, the output should be some constant, but in fact, x is the independent variable of the ode and f itself is a set of data. Which you want to integrate over the domain of the ode. This can be done like this,

data = Table[{y, y + Sin[2 Pi y]}, {y, 0, 4.82, 0.1}];
f = Interpolation[data];
sol = NDSolve[{Integrate[f[y]*ϕ[x], {y, 0, x}] + ϕ'[x] == 
    1, ϕ[0] == 0}, ϕ, {x, 0, 4}]
Plot[Evaluate[ϕ[x] /. sol], {x, 0, 4}]