I have encountered a problem during my thesis study. I need to find $F(x)$ in terms of $G(y)$. My statement as follows: $$\int_{0}^\infty F(x)[x^3*B*J_0(xy)+x^4*J_1(xy)]dx=G(y)$$ where $B$ is a constant, $G(y)$ is an unknown function of $(n-1)$. degree polynomial, 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: $$x^3*F(x)=\int_{0}^\infty G(y) *J_1(xy)*y*dy$$ However $B$ is not zero, then How can I represent $F(x)$ in terms of $G(y)$? I want to obtain only $F$ at the left side of the equation, there can be anything at the right, derivative or integral of $G$. Thanks.

I have encountered a problem during my thesis study. I need to find $F(x)$ in terms of $G(y)$. My statement as follows: $$\int_{0}^\infty F(x)[Bx^3J_0(xy)+x^4J_1(xy)]dx=G(y)$$ where $B$ is a constant, $G(y)$ is an unknown function of $(n-1)$. degree polynomial, and $J_0$ and $J_1$ are the Bessel functions of the first kind. If $B=0$, F(x) could be found using Hankel transform as follows: $$x^3F(x)=\int_{0}^\infty yG(y) J_1(xy)\,dy$$

Question. However, if $B\neq0$ then how can I represent $F(x)$ in terms of $G(y)$? I want to obtain only $F$ an the left side of the equation, there can be anything at the right, derivative or integral of $G$.

Thanks.

I have encountered a problem during my thesis study. I need to find $F(x)$ in terms of $G(y)$. My statement as follows: $$\int_{0}^\infty F(x)[x^3*B*J_0(xy)+x^4*J_1(xy)]dx=G(y)$$ where $B$ is a constant, $G(y)$ is an unknown function of $(n-1)$. degree polynomial, 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: $$x^3*F(x)=\int_{0}^\infty G(y) *J_1(xy)*y*dy$$ However $B$ is not zero, then How can I represent $F(x)$ in terms of $G(y)$? Thanks.

I have encountered a problem during my thesis study. I need to find $F(x)$ in terms of $G(y)$. My statement as follows: $$\int_{0}^\infty F(x)[x^3*B*J_0(xy)+x^4*J_1(xy)]dx=G(y)$$ where $B$ is a constant, $G(y)$ is an unknown function of $(n-1)$. degree polynomial, 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: $$x^3*F(x)=\int_{0}^\infty G(y) *J_1(xy)*y*dy$$ However $B$ is not zero, then How can I represent $F(x)$ in terms of $G(y)$? I want to obtain only $F$ at the left side of the equation, there can be anything at the right, derivative or integral of $G$. Thanks.