Find the general solution of
\begin{equation*}
x^4 y^{(4)} + 6 x^3 y^{(3)} + 7 x^2 y^{(2)} + x y' - y = 3 \ln x + \cos (\ln x) \, .
\end{equation*}
Solution
It is Euler's equation. Its characteristic polynomial is
\begin{equation*}
r(r-1)(r-2)(r-3)+6r(r-1)(r-2) +7r(r-1)+r -1=\\
r^4 - 6r^3 +11r^2 -6r + 6r^3 -18r^2 +12r +7r^2-7r+r-1=
r^4-1
\end{equation*}
with characteristic roots $r_{1,2}=\pm 1$, $r_{3,4}=\pm i$ and plugging $t=\ln x$ we arrive to
\begin{equation}
y^{(4)}_t-y= 3t + \cos (t).
\label{eq-2-1}
\end{equation}
Solution to homogeneous equation is
\begin{equation}
z= C_1 e^t +C_2 e^{-t}+C_3 \cos (t)+ C_4\sin (t)=\\
C _1 x +C_2 x^{-1}+ C_3\cos (\ln x) + C_4\sin(\ln x)
\label{eq-2-2}
\end{equation}
and the particular solution to inhomogeneous equation is $y_p = y_{p1}+y_{p2}$ with $y_{p1}= at +b$ and $y_{p2}= (c\cos (t) +d \sin(t))t$ solving equation with right hand expressions $f_1=3\ln x$ and $f_2= \cos(\ln x)$ respectively.
Plugging $y_{p1}$ we get
\begin{equation}
-at -b=3t \implies a=-3,b=0\implies y_{p1}=-3t=-3\ln x
\label{2-3}
\end{equation}
and plugging $y_{p2}$ we get
\begin{equation}
3(c\sin (t)-d\cos (t)) = \cos(t) \implies c=0,d=-\frac{1}{3}\implies \\
y_{p2}=-\frac{1}{3}\sin (t) t=-\frac{1}{3}\sin (\ln x)\ln x.
\label{eq-2-4}
\end{equation}
Adding (\ref{eq-2-2})--(\ref{eq-2-4}) we get
\begin{equation*}
y= C _1 x +C_2 x^{-1}+ C_3\cos (\ln x) + C_4\sin(\ln x)-3\ln x -\frac{1}{3}\sin (\ln x)\ln x.
\end{equation*}
