Consider $$ \frac{dy}{dx} + p(x)y = \int_{0}^{\infty} y(x)dx. $$
I want to solve above differential equation. Can I consider right hand side as constant to solve this?
I know RHS is a constant but it also involves solution $y$, which might create trouble unless solution is known to us.
Also is it possible to find solution to given ordinary differential equation which is independent of $y$.
You are correct in considering the right side first as a mostly independent constant. Then you can apply the integration formula $$ y'+py=C\implies y(x)=y(0)e^{-P(x)}+C\int_0^xe^{P(s)-P(x)}ds. $$
Now with this you can return to the original equation to try to determine $C$ $$ C=\int_0^\infty y(x)\,dx=y(0)\int_0^\infty e^{-P(x)}\,dx+C\int_0^\infty\int_0^xe^{P(s)-P(x)}ds\,dx $$ Now if the integrals involved have finite values, and if then $C$ does not cancel from the equation, you can get a value for $C$ that in addition is proportional to $y(0)$ (as the linearity of the original equation demands).