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I am an engineering student. I have encountered a problem during my thesis study. I need to find $F(x)$ in terms of $G(y)$ which is a function of $(n-1)$. degree polynomial. My statement as follows: $$\int_{0}^\infty F(x)[B\,x^3J_0(xy)+x^4J_1(xy)]dx=G(y)$$ where B is a constant, and $J_0$ and $J_1$ are the Bessel functions of the first kind. If $B$ was equal to zero, $F(x)$ could be found using Hankel transform as follows: $$F(x)=\frac{1}{x^3} \int_{0}^\infty G(y) J_1(xy)y\,dy$$ However $B$ is not zero, therefore I could not take Hankel Transform directly. Is there any approximate solution in which I can represent $F(x)$ in terms of $G(y)$ such that $$F(x)=H(x) \int_{0}^\infty G(y) J_1(xy)y\,dy$$ Thanks.

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