I was wondering if it is actually possible to find a particular solution to the following equation $$ \frac{d^2 u}{dt^2} - u =\ln t$$
I made an attempt by varying constants of the homogeneous equation's general solution but was not able to solve a final integral.
Could someone help on this question? Thanks
By using variation of parameters we find a particular solution of the form $$u(t)=A(t)e^t+B(t)e^{-t}$$ where, unfortunately, functions $A$ and $B$ are not elementary: $$\begin{cases} A'(t)e^t+B'(t)e^{-t}=0\\ A'(t)e^t-B'(t)e^{-t}=\ln(t) \end{cases}\implies \begin{cases} 2A'(t)=e^{-t}\ln(t)\\ 2B'(t)=-e^{t}\ln(t) \end{cases}$$ and therefore $$2A(t)=\int e^{-t}\ln(t)dt=-e^{-t}\ln(t)+\int \frac{e^{-t}}{t}\,dt$$ and $$2B(t)=\int e^{t}\ln(t)dt=e^{t}\ln(t)-\int \frac{e^{t}}{t}\,dt$$ which are related to the Exponential Integral $\text{Ei}(t)$ (see also WA).