Solving integral equations of the form $\int_0^1 f(t)J_\nu(\lambda t)\,\mathrm{d}t = F(\lambda)$. The goal is to solve analytically or numerically the following two integral functions for the two unknowns $f_1(t)$ and $f_2(t)$:
\begin{align}
\int_0^1 f_1(t) J_{3/2} (\lambda t) \, \mathrm{d} t = F_1(\lambda) \, , \\
\int_0^1 f_2(t) J_{1/2} (\lambda t) \, \mathrm{d} t = F_2(\lambda) \, , 
\end{align}
where the right-hand sides are explicitly given by
\begin{align}
F_1(\lambda) &= \frac{2\lambda^{3/2}e^{-\lambda/2} \left( (7\lambda+3) e^{-\lambda}-1 \right)}{1+2\lambda e^{-\lambda} - e^{-2\lambda}} \, , \\
F_2(\lambda) &= \frac{2\lambda^{1/2} e^{-\lambda/2} \left( (7\lambda^2-4\lambda-4)e^{-\lambda} + 4 + 8\lambda \right)}{1+2\lambda e^{-\lambda} - e^{-2\lambda}} \, .
\end{align}
Here, $\lambda \in [0,\infty)$. 
i have tried the standard approach of expressing the solutions as 
$$
f_1(t) = \sum_{k=0}^N a_kt^k \, , \qquad
f_2(t) = \sum_{k=0}^N b_kt^k \, , 
$$
ans solving numerically the resulting linear system of equations for the unknown coefficients $a_k$ and $b_k$, $k = 0, \dots, N$.
However, the resulting coefficients are very large and the solution is strongly oscillating.
i was wondering whether an alternative procedure can be employed in order to obtain solutions for the above integral equations.
Any help or hints are highly appreciated and desirable.
Thank you
 A: Use the Bessel orthogonality integral
$$
\int_0^{\infty} d\lambda\; \lambda\; J_{\nu}(\lambda t)\; J_{\nu}(\lambda u) \;=\; \frac{1}{u} \, \delta(t - u)\, .
$$
Multiply both sides of your equation by $\lambda\; J_{\nu}(\lambda u)$, where $u \in (0,1)$, and integrate from $\lambda = 0$ to $\lambda = \infty$ to yield:
$$
\int_0^{\infty} d\lambda\; \lambda\; J_{\nu}(\lambda u) \; \int_0^1 dt\; f(t)\; J_{\nu}(\lambda t) \;=\; \int_0^{\infty} d\lambda\; \lambda\; J_{\nu}(\lambda u)\; F(\lambda)
$$
Now assume that one can exchange the order of integration on the left hand side:
$$
\int_0^1 dt\; f(t)\; \int_0^{\infty} d\lambda\; \lambda\; J_{\nu}(\lambda u)\;J_{\nu}(\lambda t)
\;=\;
\int_0^1 dt\; f(t)\;\frac{1}{u} \, \delta(t - u) 
\;=\;
\frac{f(u)}{u}
$$
We are finally left with $f(u)$ as an integral:
$$
f(u) \;=\; u \int_0^{\infty} d\lambda\; \lambda\; J_{\nu}(\lambda u)\; F(\lambda)
$$
I leave it to you to attempt this integral for your two cases...
Edited to add: For whatever it's worth, I find that Mathematica cannot solve the two resulting integrals, which doesn't necessarily mean they don't have a closed form, just that if they do it won't be straightforward.
