I am try to solve the following Problem
the nonhomogeneous term contains modified bessel function K1
please give me advice
$$\lambda^2 Y- Y''= \frac{\lambda^3 (G) K[1,\lambda(G^2+y^2)^{1/2})]}{(G^2+y^2)^{1/2}}$$
I tried
DSolve [\[Lambda]^2 f[y] - f''[y] == (\[Lambda]^3 G BesselK[1, \[Lambda] Sqrt[y^2 + G^2]])/Sqrt[y^2 + G^2], f[y], y]
Try variation of parameters.
But first replace the 1-variable 2nd-order equation with 2-variable 1st-order equations (I assume you're familiar with matrix notation) :
$$\begin{bmatrix} Y' \\ V' \end{bmatrix} = \begin{bmatrix} V \\ \lambda^2Y-\lambda^3G\,K_1\Big(\lambda\sqrt{G^2+y^2}\Big)/\sqrt{G^2+y^2} \end{bmatrix}.$$
From Adrian's comment, the homogeneous part of this,
$$\begin{bmatrix} Y' \\ V' \end{bmatrix} = \begin{bmatrix} V \\ \lambda^2Y \end{bmatrix},$$
has solutions
$$\begin{bmatrix} Y \\ V \end{bmatrix} = c_1\begin{bmatrix} e^{\lambda y} \\ \lambda e^{\lambda y} \end{bmatrix} + c_2\begin{bmatrix} e^{-\lambda y} \\ -\lambda e^{-\lambda y} \end{bmatrix}.$$
Now replace $c_1$ and $c_2$ with variables $u_1$ and $u_2$, and the original equation reduces to
$$u_1'\begin{bmatrix} e^{\lambda y} \\ \lambda e^{\lambda y} \end{bmatrix} + u_2'\begin{bmatrix} e^{-\lambda y} \\ -\lambda e^{-\lambda y} \end{bmatrix} = \begin{bmatrix} 0 \\ -\lambda^3G\,K_1\Big(\lambda\sqrt{G^2+y^2}\Big)/\sqrt{G^2+y^2} \end{bmatrix},$$
which gives
$$u_1'e^{\lambda y}=-u_2'e^{-\lambda y}=-\lambda^2G\frac{K_1\Big(\lambda\sqrt{G^2+y^2}\Big)}{2\sqrt{G^2+y^2}}.$$
I don't know if this helps; I don't have much experience with Bessel functions.