8.1.3.95 Higher order differential equations

Problem: Find the general solution of the given equations, knowing their particular solutions. In those problems where the particular solution is not given, we can look for it by selecting, for example, in the form of an exponential function \( y_{1}=e^{a x} \) or algebraic polynomial \( y_{1}=x^{n}+a x^{n-1}+b x^{n-2}+\cdots \). \[ \begin{array}{l} \left(x^{2}-2 x+3\right) y^{\prime \prime \prime}-\left(x^{2}+1\right) y^{\prime \prime}+2 x y^{\prime}-2 y=0, \\ y_{1}=x, y_{2}=e^{x} . \end{array} \]