I want to solve this differential equation evaluted at $x=0$. I will start with an example with $m=2$
$$ f(0)+f'(0)+f''(0)=2\lambda_{1}\lambda_{2}+4 $$ where $m$,$\lambda_{1}$ and $\lambda_{1}$ are positive constants. The $f(x)$ that satisfies the previous equation is
$$ \prod_{k=1}^{2}\left(\lambda_{k}.x+2(1-\lambda_{k})\right) $$
I have encountered with this equation while solving an integral. The result was
$$ \left. \sum_{n=0}^{m+1} \left( \frac{d^n}{d x^n}\left( x^{m-1}\prod_{k=1}^{2}\left(\lambda_{k}.x+m(1-\lambda_{k})\right)\right)\right)\right|_{x=0}=m\lambda_{1}\lambda_{2}+m^2 $$
So I want to find f(x) that satisfies
$$ \left. \sum_{n=0}^{m+1} \left( \frac{d^n}{d x^n}\left( f(x) \right)\right)\right|_{x=0}=m_{1}\lambda_{1}\lambda_{2}+m_{1}m_{2} $$
Note: $m_{1}\neq m_{2}$.