This article concerns the problem on the growth and the oscillation of some differential polynomials generated by solutions of the second order non-homogeneous linear differential equation \[\begin{equation*} f^{\prime \prime }+P\left( z\right) e^{a_{n}z^{n}}f^{\prime }+B\left( z\right) e^{b_{n}z^{n}}f=F\left( z\right) e^{a_{n}z^{n}}, \end{equation*}\] where \(a_{n}\), \(b_{n}\) are complex numbers, \(P\left( z\right)\) \(\left( \not\equiv 0\right)\) is a polynomial, \(B\left( z\right)\) \(\left( \not\equiv 0\right)\) and \(F\left( z\right)\) \(\left( \not\equiv 0\right)\) are entire functions with order less than \(n\). Because of the control of differential equation, we can obtain some estimates of their hyper-order and fixed points.
