problem 1517

Internal problem ID [9841]
Internal ﬁle name [OUTPUT/8786_Monday_June_06_2022_05_27_50_AM_75423132/index.tex]

Book: Diﬀerential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 3, linear third order
Problem number: 1517.
ODE order: 3.
ODE degree: 1.

The type(s) of ODE detected by this program : "higher_order_ODE_non_constant_coeﬃcients_of_type_Euler"

\[ \boxed {y^{\prime \prime \prime } x^{3}+x^{2} y^{\prime \prime }+2 y^{\prime } x -y=2 x^{3}-\ln \left (x \right )} \] This is higher order nonhomogeneous Euler type ODE. Let the solution be \[ y = y_h + y_p \] Where $y_h$ is the solution to the homogeneous Euler ODE And $y_p$ is a particular solution to the nonhomogeneous Euler ODE. $y_h$ is the solution to \[ y^{\prime \prime \prime } x^{3}+x^{2} y^{\prime \prime }+2 y^{\prime } x -y = 0 \] This is Euler ODE of higher order. Let $y = x^{\lambda }$. Hence \begin {align*} y^{\prime } &= \lambda \,x^{\lambda -1}\\ y^{\prime \prime } &= \lambda \left (\lambda -1\right ) x^{\lambda -2}\\ y^{\prime \prime \prime } &= \lambda \left (\lambda -1\right ) \left (\lambda -2\right ) x^{\lambda -3} \end {align*}

Substituting these back into \[ y^{\prime \prime \prime } x^{3}+x^{2} y^{\prime \prime }+2 y^{\prime } x -y = 2 x^{3}-\ln \left (x \right ) \] gives \[ 2 x \lambda \,x^{\lambda -1}+x^{2} \lambda \left (\lambda -1\right ) x^{\lambda -2}+x^{3} \lambda \left (\lambda -1\right ) \left (\lambda -2\right ) x^{\lambda -3}-x^{\lambda } = 0 \] Which simpliﬁes to \[ 2 \lambda \,x^{\lambda }+\lambda \left (\lambda -1\right ) x^{\lambda }+\lambda \left (\lambda -1\right ) \left (\lambda -2\right ) x^{\lambda }-x^{\lambda } = 0 \] And since $x^{\lambda }\neq 0$ then dividing through by $x^{\lambda }$, the above becomes

\[ 2 \lambda +\lambda \left (\lambda -1\right )+\lambda \left (\lambda -1\right ) \left (\lambda -2\right )-1 = 0 \] Simplifying gives the characteristic equation as \[ \lambda ^{3}-2 \lambda ^{2}+3 \lambda -1 = 0 \] Solving the above gives the following roots \begin {align*} \lambda _1 &= -\frac {\left (44+12 \sqrt {69}\right )^{\frac {1}{3}}}{6}+\frac {10}{3 \left (44+12 \sqrt {69}\right )^{\frac {1}{3}}}+\frac {2}{3}\\ \lambda _2 &= \frac {\left (44+12 \sqrt {69}\right )^{\frac {1}{3}}}{12}-\frac {5}{3 \left (44+12 \sqrt {69}\right )^{\frac {1}{3}}}+\frac {2}{3}+\frac {i \sqrt {3}\, \left (-\frac {\left (44+12 \sqrt {69}\right )^{\frac {1}{3}}}{6}-\frac {10}{3 \left (44+12 \sqrt {69}\right )^{\frac {1}{3}}}\right )}{2}\\ \lambda _3 &= \frac {\left (44+12 \sqrt {69}\right )^{\frac {1}{3}}}{12}-\frac {5}{3 \left (44+12 \sqrt {69}\right )^{\frac {1}{3}}}+\frac {2}{3}-\frac {i \sqrt {3}\, \left (-\frac {\left (44+12 \sqrt {69}\right )^{\frac {1}{3}}}{6}-\frac {10}{3 \left (44+12 \sqrt {69}\right )^{\frac {1}{3}}}\right )}{2} \end {align*}


root	multiplicity	type of root

$\frac {\left (44+12 \sqrt {69}\right )^{\frac {1}{3}}}{12}-\frac {5}{3 \left (44+12 \sqrt {69}\right )^{\frac {1}{3}}}+\frac {2}{3} \pm -\frac {\sqrt {3}\, \left (-\frac {\left (44+12 \sqrt {69}\right )^{\frac {1}{3}}}{6}-\frac {10}{3 \left (44+12 \sqrt {69}\right )^{\frac {1}{3}}}\right )}{2} i$	$1$	complex conjugate root

$-\frac {\left (44+12 \sqrt {69}\right )^{\frac {1}{3}}}{6}+\frac {10}{3 \left (44+12 \sqrt {69}\right )^{\frac {1}{3}}}+\frac {2}{3}$	$1$	real root

The solution is generated by going over the above table. For each real root $\lambda $ of multiplicity one generates a $c_1x^{\lambda }$ basis solution. Each real root of multiplicty two, generates $c_1x^{\lambda }$ and $c_2x^{\lambda } \ln \left (x \right )$ basis solutions. Each real root of multiplicty three, generates $c_1x^{\lambda }$ and $c_2x^{\lambda } \ln \left (x \right )$ and $c_3x^{\lambda } \ln \left (x \right )^{2}$ basis solutions, and so on. Each complex root $\alpha \pm i \beta $ of multiplicity one generates $x^{\alpha } \left (c_1\cos (\beta \ln \left (x \right ))+c_2\sin (\beta \ln \left (x \right ))\right )$ basis solutions. And each complex root $\alpha \pm i \beta $ of multiplicity two generates $\ln \left (x \right ) x^{\alpha }\left (c_1\cos (\beta \ln \left (x \right ))+c_2\sin (\beta \ln \left (x \right ))\right )$ basis solutions. And each complex root $\alpha \pm i \beta $ of multiplicity three generates $\ln \left (x \right )^{2} x^{\alpha }\left (c_1\cos (\beta \ln \left (x \right ))+c_2\sin (\beta \ln \left (x \right ))\right )$ basis solutions. And so on. Using the above show that the solution is

\[ y = x^{\frac {\left (44+12 \sqrt {69}\right )^{\frac {1}{3}}}{12}-\frac {5}{3 \left (44+12 \sqrt {69}\right )^{\frac {1}{3}}}+\frac {2}{3}} \left (c_{1} \cos \left (\frac {\sqrt {3}\, \left (-\frac {\left (44+12 \sqrt {69}\right )^{\frac {1}{3}}}{6}-\frac {10}{3 \left (44+12 \sqrt {69}\right )^{\frac {1}{3}}}\right ) \ln \left (x \right )}{2}\right )-c_{2} \sin \left (\frac {\sqrt {3}\, \left (-\frac {\left (44+12 \sqrt {69}\right )^{\frac {1}{3}}}{6}-\frac {10}{3 \left (44+12 \sqrt {69}\right )^{\frac {1}{3}}}\right ) \ln \left (x \right )}{2}\right )\right )+c_{3} x^{-\frac {\left (44+12 \sqrt {69}\right )^{\frac {1}{3}}}{6}+\frac {10}{3 \left (44+12 \sqrt {69}\right )^{\frac {1}{3}}}+\frac {2}{3}} \] The fundamental set of solutions for the homogeneous solution are the following \begin{align*} y_1 &= x^{\frac {\left (44+12 \sqrt {69}\right )^{\frac {1}{3}}}{12}-\frac {5}{3 \left (44+12 \sqrt {69}\right )^{\frac {1}{3}}}+\frac {2}{3}} \cos \left (\frac {\sqrt {3}\, \left (-\frac {\left (44+12 \sqrt {69}\right )^{\frac {1}{3}}}{6}-\frac {10}{3 \left (44+12 \sqrt {69}\right )^{\frac {1}{3}}}\right ) \ln \left (x \right )}{2}\right ) \\ y_2 &= -x^{\frac {\left (44+12 \sqrt {69}\right )^{\frac {1}{3}}}{12}-\frac {5}{3 \left (44+12 \sqrt {69}\right )^{\frac {1}{3}}}+\frac {2}{3}} \sin \left (\frac {\sqrt {3}\, \left (-\frac {\left (44+12 \sqrt {69}\right )^{\frac {1}{3}}}{6}-\frac {10}{3 \left (44+12 \sqrt {69}\right )^{\frac {1}{3}}}\right ) \ln \left (x \right )}{2}\right ) \\ y_3 &= x^{-\frac {\left (44+12 \sqrt {69}\right )^{\frac {1}{3}}}{6}+\frac {10}{3 \left (44+12 \sqrt {69}\right )^{\frac {1}{3}}}+\frac {2}{3}} \\ \end{align*} Now the particular solution to the given ODE is found \[ y^{\prime \prime \prime } x^{3}+x^{2} y^{\prime \prime }+2 y^{\prime } x -y = 2 x^{3}-\ln \left (x \right ) \] Let the particular solution be \[ y_p = U_1 y_1+U_2 y_2+U_3 y_3 \] Where $y_i$ are the basis solutions found above for the homogeneous solution $y_h$ and $U_i(x)$ are functions to be determined as follows \[ U_i = (-1)^{n-i} \int { \frac {F(x) W_i(x) }{a W(x)} \, dx} \] Where $W(x)$ is the Wronskian and $W_i(x)$ is the Wronskian that results after deleting the last row and the $i$-th column of the determinant and $n$ is the order of the ODE or equivalently, the number of basis solutions, and $a$ is the coeﬃcient of the leading derivative in the ODE, and $F(x)$ is the RHS of the ODE. Therefore, the ﬁrst step is to ﬁnd the Wronskian $W \left (x \right )$. This is given by \begin {equation*} W(x) = \begin {vmatrix} y_1&y_2&y_3\\ y_1'&y_2'&y_3'\\ y_1''&y_2''&y_3''\\ \end {vmatrix} \end {equation*} Substituting the fundamental set of solutions $y_i$ found above in the Wronskian gives \begin {align*} W &= \text {Expression too large to display} \\ |W| &= \text {Expression too large to display} \end {align*}

The determinant simpliﬁes to \begin {align*} |W| &= \frac {\sqrt {23}}{2 x} \end {align*}

Now we determine $W_i$ for each $U_i$. \begin {align*} W_1(x) &= \det \,\left [\begin {array}{cc} x^{\frac {\left (\left (44+12 \sqrt {69}\right )^{\frac {1}{3}}+10\right ) \left (\left (44+12 \sqrt {69}\right )^{\frac {1}{3}}-2\right )}{12 \left (44+12 \sqrt {69}\right )^{\frac {1}{3}}}} \sin \left (\frac {\sqrt {3}\, \left (\left (44+12 \sqrt {3}\, \sqrt {23}\right )^{\frac {2}{3}}+20\right ) \ln \left (x \right )}{12 \left (44+12 \sqrt {3}\, \sqrt {23}\right )^{\frac {1}{3}}}\right ) & x^{-\frac {\left (44+12 \sqrt {69}\right )^{\frac {1}{3}}}{6}+\frac {10}{3 \left (44+12 \sqrt {69}\right )^{\frac {1}{3}}}+\frac {2}{3}} \\ \frac {\left (\left (\left (44+12 \sqrt {3}\, \sqrt {23}\right )^{\frac {2}{3}}+8 \left (44+12 \sqrt {3}\, \sqrt {23}\right )^{\frac {1}{3}}-20\right ) \sin \left (\frac {\sqrt {3}\, \left (\left (44+12 \sqrt {3}\, \sqrt {23}\right )^{\frac {2}{3}}+20\right ) \ln \left (x \right )}{12 \left (44+12 \sqrt {3}\, \sqrt {23}\right )^{\frac {1}{3}}}\right )+\sqrt {3}\, \left (\left (44+12 \sqrt {3}\, \sqrt {23}\right )^{\frac {2}{3}}+20\right ) \cos \left (\frac {\sqrt {3}\, \left (\left (44+12 \sqrt {3}\, \sqrt {23}\right )^{\frac {2}{3}}+20\right ) \ln \left (x \right )}{12 \left (44+12 \sqrt {3}\, \sqrt {23}\right )^{\frac {1}{3}}}\right )\right ) x^{-\frac {-\frac {\left (44+12 \sqrt {3}\, \sqrt {23}\right )^{\frac {2}{3}}}{4}+\left (44+12 \sqrt {3}\, \sqrt {23}\right )^{\frac {1}{3}}+5}{3 \left (44+12 \sqrt {3}\, \sqrt {23}\right )^{\frac {1}{3}}}}}{12 \left (44+12 \sqrt {3}\, \sqrt {23}\right )^{\frac {1}{3}}} & -\frac {x^{-\frac {\left (44+12 \sqrt {69}\right )^{\frac {2}{3}}+2 \left (44+12 \sqrt {69}\right )^{\frac {1}{3}}-20}{6 \left (44+12 \sqrt {69}\right )^{\frac {1}{3}}}} \left (\left (44+12 \sqrt {69}\right )^{\frac {2}{3}}-4 \left (44+12 \sqrt {69}\right )^{\frac {1}{3}}-20\right )}{6 \left (44+12 \sqrt {69}\right )^{\frac {1}{3}}} \end {array}\right ] \\ &= -\frac {3 x^{-\frac {\left (44+12 \sqrt {3}\, \sqrt {23}\right )^{\frac {2}{3}}-4 \left (44+12 \sqrt {3}\, \sqrt {23}\right )^{\frac {1}{3}}-20}{12 \left (44+12 \sqrt {3}\, \sqrt {23}\right )^{\frac {1}{3}}}} \left (\left (\frac {5 \sqrt {3}\, \left (44+12 \sqrt {3}\, \sqrt {23}\right )^{\frac {1}{3}}}{9}+\frac {11 \sqrt {3}}{9}+\sqrt {23}\right ) \cos \left (\frac {\sqrt {3}\, \left (\left (44+12 \sqrt {3}\, \sqrt {23}\right )^{\frac {2}{3}}+20\right ) \ln \left (x \right )}{12 \left (44+12 \sqrt {3}\, \sqrt {23}\right )^{\frac {1}{3}}}\right )+\sin \left (\frac {\sqrt {3}\, \left (\left (44+12 \sqrt {3}\, \sqrt {23}\right )^{\frac {2}{3}}+20\right ) \ln \left (x \right )}{12 \left (44+12 \sqrt {3}\, \sqrt {23}\right )^{\frac {1}{3}}}\right ) \left (\sqrt {3}\, \sqrt {23}-\frac {5 \left (44+12 \sqrt {3}\, \sqrt {23}\right )^{\frac {1}{3}}}{3}+\frac {11}{3}\right )\right )}{\left (44+12 \sqrt {3}\, \sqrt {23}\right )^{\frac {2}{3}}} \end {align*}

\begin {align*} W_2(x) &= \det \,\left [\begin {array}{cc} x^{\frac {\left (\left (44+12 \sqrt {69}\right )^{\frac {1}{3}}+10\right ) \left (\left (44+12 \sqrt {69}\right )^{\frac {1}{3}}-2\right )}{12 \left (44+12 \sqrt {69}\right )^{\frac {1}{3}}}} \cos \left (\frac {\sqrt {3}\, \left (\left (44+12 \sqrt {3}\, \sqrt {23}\right )^{\frac {2}{3}}+20\right ) \ln \left (x \right )}{12 \left (44+12 \sqrt {3}\, \sqrt {23}\right )^{\frac {1}{3}}}\right ) & x^{-\frac {\left (44+12 \sqrt {69}\right )^{\frac {1}{3}}}{6}+\frac {10}{3 \left (44+12 \sqrt {69}\right )^{\frac {1}{3}}}+\frac {2}{3}} \\ -\frac {\left (\left (-\left (44+12 \sqrt {3}\, \sqrt {23}\right )^{\frac {2}{3}}-8 \left (44+12 \sqrt {3}\, \sqrt {23}\right )^{\frac {1}{3}}+20\right ) \cos \left (\frac {\sqrt {3}\, \left (\left (44+12 \sqrt {3}\, \sqrt {23}\right )^{\frac {2}{3}}+20\right ) \ln \left (x \right )}{12 \left (44+12 \sqrt {3}\, \sqrt {23}\right )^{\frac {1}{3}}}\right )+\sqrt {3}\, \left (\left (44+12 \sqrt {3}\, \sqrt {23}\right )^{\frac {2}{3}}+20\right ) \sin \left (\frac {\sqrt {3}\, \left (\left (44+12 \sqrt {3}\, \sqrt {23}\right )^{\frac {2}{3}}+20\right ) \ln \left (x \right )}{12 \left (44+12 \sqrt {3}\, \sqrt {23}\right )^{\frac {1}{3}}}\right )\right ) x^{-\frac {-\frac {\left (44+12 \sqrt {3}\, \sqrt {23}\right )^{\frac {2}{3}}}{4}+\left (44+12 \sqrt {3}\, \sqrt {23}\right )^{\frac {1}{3}}+5}{3 \left (44+12 \sqrt {3}\, \sqrt {23}\right )^{\frac {1}{3}}}}}{12 \left (44+12 \sqrt {3}\, \sqrt {23}\right )^{\frac {1}{3}}} & -\frac {x^{-\frac {\left (44+12 \sqrt {69}\right )^{\frac {2}{3}}+2 \left (44+12 \sqrt {69}\right )^{\frac {1}{3}}-20}{6 \left (44+12 \sqrt {69}\right )^{\frac {1}{3}}}} \left (\left (44+12 \sqrt {69}\right )^{\frac {2}{3}}-4 \left (44+12 \sqrt {69}\right )^{\frac {1}{3}}-20\right )}{6 \left (44+12 \sqrt {69}\right )^{\frac {1}{3}}} \end {array}\right ] \\ &= \frac {5 x^{-\frac {\left (44+12 \sqrt {3}\, \sqrt {23}\right )^{\frac {2}{3}}-4 \left (44+12 \sqrt {3}\, \sqrt {23}\right )^{\frac {1}{3}}-20}{12 \left (44+12 \sqrt {3}\, \sqrt {23}\right )^{\frac {1}{3}}}} \left (\left (-\frac {9 \sqrt {3}\, \sqrt {23}}{5}+3 \left (44+12 \sqrt {3}\, \sqrt {23}\right )^{\frac {1}{3}}-\frac {33}{5}\right ) \cos \left (\frac {\sqrt {3}\, \left (\left (44+12 \sqrt {3}\, \sqrt {23}\right )^{\frac {2}{3}}+20\right ) \ln \left (x \right )}{12 \left (44+12 \sqrt {3}\, \sqrt {23}\right )^{\frac {1}{3}}}\right )+\sin \left (\frac {\sqrt {3}\, \left (\left (44+12 \sqrt {3}\, \sqrt {23}\right )^{\frac {2}{3}}+20\right ) \ln \left (x \right )}{12 \left (44+12 \sqrt {3}\, \sqrt {23}\right )^{\frac {1}{3}}}\right ) \left (\sqrt {3}\, \left (44+12 \sqrt {3}\, \sqrt {23}\right )^{\frac {1}{3}}+\frac {11 \sqrt {3}}{5}+\frac {9 \sqrt {23}}{5}\right )\right )}{3 \left (44+12 \sqrt {3}\, \sqrt {23}\right )^{\frac {2}{3}}} \end {align*}

\begin {align*} W_3(x) &= \det \,\left [\begin {array}{cc} x^{\frac {\left (\left (44+12 \sqrt {69}\right )^{\frac {1}{3}}+10\right ) \left (\left (44+12 \sqrt {69}\right )^{\frac {1}{3}}-2\right )}{12 \left (44+12 \sqrt {69}\right )^{\frac {1}{3}}}} \cos \left (\frac {\sqrt {3}\, \left (\left (44+12 \sqrt {3}\, \sqrt {23}\right )^{\frac {2}{3}}+20\right ) \ln \left (x \right )}{12 \left (44+12 \sqrt {3}\, \sqrt {23}\right )^{\frac {1}{3}}}\right ) & x^{\frac {\left (\left (44+12 \sqrt {69}\right )^{\frac {1}{3}}+10\right ) \left (\left (44+12 \sqrt {69}\right )^{\frac {1}{3}}-2\right )}{12 \left (44+12 \sqrt {69}\right )^{\frac {1}{3}}}} \sin \left (\frac {\sqrt {3}\, \left (\left (44+12 \sqrt {3}\, \sqrt {23}\right )^{\frac {2}{3}}+20\right ) \ln \left (x \right )}{12 \left (44+12 \sqrt {3}\, \sqrt {23}\right )^{\frac {1}{3}}}\right ) \\ -\frac {\left (\left (-\left (44+12 \sqrt {3}\, \sqrt {23}\right )^{\frac {2}{3}}-8 \left (44+12 \sqrt {3}\, \sqrt {23}\right )^{\frac {1}{3}}+20\right ) \cos \left (\frac {\sqrt {3}\, \left (\left (44+12 \sqrt {3}\, \sqrt {23}\right )^{\frac {2}{3}}+20\right ) \ln \left (x \right )}{12 \left (44+12 \sqrt {3}\, \sqrt {23}\right )^{\frac {1}{3}}}\right )+\sqrt {3}\, \left (\left (44+12 \sqrt {3}\, \sqrt {23}\right )^{\frac {2}{3}}+20\right ) \sin \left (\frac {\sqrt {3}\, \left (\left (44+12 \sqrt {3}\, \sqrt {23}\right )^{\frac {2}{3}}+20\right ) \ln \left (x \right )}{12 \left (44+12 \sqrt {3}\, \sqrt {23}\right )^{\frac {1}{3}}}\right )\right ) x^{-\frac {-\frac {\left (44+12 \sqrt {3}\, \sqrt {23}\right )^{\frac {2}{3}}}{4}+\left (44+12 \sqrt {3}\, \sqrt {23}\right )^{\frac {1}{3}}+5}{3 \left (44+12 \sqrt {3}\, \sqrt {23}\right )^{\frac {1}{3}}}}}{12 \left (44+12 \sqrt {3}\, \sqrt {23}\right )^{\frac {1}{3}}} & \frac {\left (\left (\left (44+12 \sqrt {3}\, \sqrt {23}\right )^{\frac {2}{3}}+8 \left (44+12 \sqrt {3}\, \sqrt {23}\right )^{\frac {1}{3}}-20\right ) \sin \left (\frac {\sqrt {3}\, \left (\left (44+12 \sqrt {3}\, \sqrt {23}\right )^{\frac {2}{3}}+20\right ) \ln \left (x \right )}{12 \left (44+12 \sqrt {3}\, \sqrt {23}\right )^{\frac {1}{3}}}\right )+\sqrt {3}\, \left (\left (44+12 \sqrt {3}\, \sqrt {23}\right )^{\frac {2}{3}}+20\right ) \cos \left (\frac {\sqrt {3}\, \left (\left (44+12 \sqrt {3}\, \sqrt {23}\right )^{\frac {2}{3}}+20\right ) \ln \left (x \right )}{12 \left (44+12 \sqrt {3}\, \sqrt {23}\right )^{\frac {1}{3}}}\right )\right ) x^{-\frac {-\frac {\left (44+12 \sqrt {3}\, \sqrt {23}\right )^{\frac {2}{3}}}{4}+\left (44+12 \sqrt {3}\, \sqrt {23}\right )^{\frac {1}{3}}+5}{3 \left (44+12 \sqrt {3}\, \sqrt {23}\right )^{\frac {1}{3}}}}}{12 \left (44+12 \sqrt {3}\, \sqrt {23}\right )^{\frac {1}{3}}} \end {array}\right ] \\ &= \frac {\sqrt {3}\, \left (\left (44+12 \sqrt {3}\, \sqrt {23}\right )^{\frac {2}{3}}+20\right ) x^{\frac {\left (44+12 \sqrt {3}\, \sqrt {23}\right )^{\frac {2}{3}}+2 \left (44+12 \sqrt {3}\, \sqrt {23}\right )^{\frac {1}{3}}-20}{6 \left (44+12 \sqrt {3}\, \sqrt {23}\right )^{\frac {1}{3}}}}}{12 \left (44+12 \sqrt {3}\, \sqrt {23}\right )^{\frac {1}{3}}} \end {align*}

Now we are ready to evaluate each $U_i(x)$. \begin {align*} U_1 &= (-1)^{3-1} \int { \frac {F(x) W_1(x) }{a W(x)} \, dx}\\ &= (-1)^{2} \int { \frac { \left (2 x^{3}-\ln \left (x \right )\right ) \left (-\frac {3 x^{-\frac {\left (44+12 \sqrt {3}\, \sqrt {23}\right )^{\frac {2}{3}}-4 \left (44+12 \sqrt {3}\, \sqrt {23}\right )^{\frac {1}{3}}-20}{12 \left (44+12 \sqrt {3}\, \sqrt {23}\right )^{\frac {1}{3}}}} \left (\left (\frac {5 \sqrt {3}\, \left (44+12 \sqrt {3}\, \sqrt {23}\right )^{\frac {1}{3}}}{9}+\frac {11 \sqrt {3}}{9}+\sqrt {23}\right ) \cos \left (\frac {\sqrt {3}\, \left (\left (44+12 \sqrt {3}\, \sqrt {23}\right )^{\frac {2}{3}}+20\right ) \ln \left (x \right )}{12 \left (44+12 \sqrt {3}\, \sqrt {23}\right )^{\frac {1}{3}}}\right )+\sin \left (\frac {\sqrt {3}\, \left (\left (44+12 \sqrt {3}\, \sqrt {23}\right )^{\frac {2}{3}}+20\right ) \ln \left (x \right )}{12 \left (44+12 \sqrt {3}\, \sqrt {23}\right )^{\frac {1}{3}}}\right ) \left (\sqrt {3}\, \sqrt {23}-\frac {5 \left (44+12 \sqrt {3}\, \sqrt {23}\right )^{\frac {1}{3}}}{3}+\frac {11}{3}\right )\right )}{\left (44+12 \sqrt {3}\, \sqrt {23}\right )^{\frac {2}{3}}}\right )}{\left (x^{3}\right ) \left (\frac {\sqrt {23}}{2 x}\right )} \, dx} \\ &= \int { \frac {-\frac {3 \left (2 x^{3}-\ln \left (x \right )\right ) x^{-\frac {\left (44+12 \sqrt {3}\, \sqrt {23}\right )^{\frac {2}{3}}-4 \left (44+12 \sqrt {3}\, \sqrt {23}\right )^{\frac {1}{3}}-20}{12 \left (44+12 \sqrt {3}\, \sqrt {23}\right )^{\frac {1}{3}}}} \left (\left (\frac {5 \sqrt {3}\, \left (44+12 \sqrt {3}\, \sqrt {23}\right )^{\frac {1}{3}}}{9}+\frac {11 \sqrt {3}}{9}+\sqrt {23}\right ) \cos \left (\frac {\sqrt {3}\, \left (\left (44+12 \sqrt {3}\, \sqrt {23}\right )^{\frac {2}{3}}+20\right ) \ln \left (x \right )}{12 \left (44+12 \sqrt {3}\, \sqrt {23}\right )^{\frac {1}{3}}}\right )+\sin \left (\frac {\sqrt {3}\, \left (\left (44+12 \sqrt {3}\, \sqrt {23}\right )^{\frac {2}{3}}+20\right ) \ln \left (x \right )}{12 \left (44+12 \sqrt {3}\, \sqrt {23}\right )^{\frac {1}{3}}}\right ) \left (\sqrt {3}\, \sqrt {23}-\frac {5 \left (44+12 \sqrt {3}\, \sqrt {23}\right )^{\frac {1}{3}}}{3}+\frac {11}{3}\right )\right )}{\left (44+12 \sqrt {3}\, \sqrt {23}\right )^{\frac {2}{3}}}}{\frac {x^{2} \sqrt {23}}{2}} \, dx}\\ &= \int {\left (-\frac {6 \left (2 x^{3}-\ln \left (x \right )\right ) x^{-\frac {\left (44+12 \sqrt {3}\, \sqrt {23}\right )^{\frac {2}{3}}-4 \left (44+12 \sqrt {3}\, \sqrt {23}\right )^{\frac {1}{3}}-20}{12 \left (44+12 \sqrt {3}\, \sqrt {23}\right )^{\frac {1}{3}}}} \left (\left (\frac {5 \sqrt {3}\, \left (44+12 \sqrt {3}\, \sqrt {23}\right )^{\frac {1}{3}}}{9}+\frac {11 \sqrt {3}}{9}+\sqrt {23}\right ) \cos \left (\frac {\sqrt {3}\, \left (\left (44+12 \sqrt {3}\, \sqrt {23}\right )^{\frac {2}{3}}+20\right ) \ln \left (x \right )}{12 \left (44+12 \sqrt {3}\, \sqrt {23}\right )^{\frac {1}{3}}}\right )+\sin \left (\frac {\sqrt {3}\, \left (\left (44+12 \sqrt {3}\, \sqrt {23}\right )^{\frac {2}{3}}+20\right ) \ln \left (x \right )}{12 \left (44+12 \sqrt {3}\, \sqrt {23}\right )^{\frac {1}{3}}}\right ) \left (\sqrt {3}\, \sqrt {23}-\frac {5 \left (44+12 \sqrt {3}\, \sqrt {23}\right )^{\frac {1}{3}}}{3}+\frac {11}{3}\right )\right ) \sqrt {23}}{23 \left (44+12 \sqrt {3}\, \sqrt {23}\right )^{\frac {2}{3}} x^{2}}\right ) \, dx}\\ &= \text {Expression too large to display} \end {align*}

\begin {align*} U_2 &= (-1)^{3-2} \int { \frac {F(x) W_2(x) }{a W(x)} \, dx}\\ &= (-1)^{1} \int { \frac { \left (2 x^{3}-\ln \left (x \right )\right ) \left (\frac {5 x^{-\frac {\left (44+12 \sqrt {3}\, \sqrt {23}\right )^{\frac {2}{3}}-4 \left (44+12 \sqrt {3}\, \sqrt {23}\right )^{\frac {1}{3}}-20}{12 \left (44+12 \sqrt {3}\, \sqrt {23}\right )^{\frac {1}{3}}}} \left (\left (-\frac {9 \sqrt {3}\, \sqrt {23}}{5}+3 \left (44+12 \sqrt {3}\, \sqrt {23}\right )^{\frac {1}{3}}-\frac {33}{5}\right ) \cos \left (\frac {\sqrt {3}\, \left (\left (44+12 \sqrt {3}\, \sqrt {23}\right )^{\frac {2}{3}}+20\right ) \ln \left (x \right )}{12 \left (44+12 \sqrt {3}\, \sqrt {23}\right )^{\frac {1}{3}}}\right )+\sin \left (\frac {\sqrt {3}\, \left (\left (44+12 \sqrt {3}\, \sqrt {23}\right )^{\frac {2}{3}}+20\right ) \ln \left (x \right )}{12 \left (44+12 \sqrt {3}\, \sqrt {23}\right )^{\frac {1}{3}}}\right ) \left (\sqrt {3}\, \left (44+12 \sqrt {3}\, \sqrt {23}\right )^{\frac {1}{3}}+\frac {11 \sqrt {3}}{5}+\frac {9 \sqrt {23}}{5}\right )\right )}{3 \left (44+12 \sqrt {3}\, \sqrt {23}\right )^{\frac {2}{3}}}\right )}{\left (x^{3}\right ) \left (\frac {\sqrt {23}}{2 x}\right )} \, dx} \\ &= - \int { \frac {\frac {5 \left (2 x^{3}-\ln \left (x \right )\right ) x^{-\frac {\left (44+12 \sqrt {3}\, \sqrt {23}\right )^{\frac {2}{3}}-4 \left (44+12 \sqrt {3}\, \sqrt {23}\right )^{\frac {1}{3}}-20}{12 \left (44+12 \sqrt {3}\, \sqrt {23}\right )^{\frac {1}{3}}}} \left (\left (-\frac {9 \sqrt {3}\, \sqrt {23}}{5}+3 \left (44+12 \sqrt {3}\, \sqrt {23}\right )^{\frac {1}{3}}-\frac {33}{5}\right ) \cos \left (\frac {\sqrt {3}\, \left (\left (44+12 \sqrt {3}\, \sqrt {23}\right )^{\frac {2}{3}}+20\right ) \ln \left (x \right )}{12 \left (44+12 \sqrt {3}\, \sqrt {23}\right )^{\frac {1}{3}}}\right )+\sin \left (\frac {\sqrt {3}\, \left (\left (44+12 \sqrt {3}\, \sqrt {23}\right )^{\frac {2}{3}}+20\right ) \ln \left (x \right )}{12 \left (44+12 \sqrt {3}\, \sqrt {23}\right )^{\frac {1}{3}}}\right ) \left (\sqrt {3}\, \left (44+12 \sqrt {3}\, \sqrt {23}\right )^{\frac {1}{3}}+\frac {11 \sqrt {3}}{5}+\frac {9 \sqrt {23}}{5}\right )\right )}{3 \left (44+12 \sqrt {3}\, \sqrt {23}\right )^{\frac {2}{3}}}}{\frac {x^{2} \sqrt {23}}{2}} \, dx}\\ &= - \int {\left (\frac {10 \left (2 x^{3}-\ln \left (x \right )\right ) x^{-\frac {\left (44+12 \sqrt {3}\, \sqrt {23}\right )^{\frac {2}{3}}-4 \left (44+12 \sqrt {3}\, \sqrt {23}\right )^{\frac {1}{3}}-20}{12 \left (44+12 \sqrt {3}\, \sqrt {23}\right )^{\frac {1}{3}}}} \left (\left (-\frac {9 \sqrt {3}\, \sqrt {23}}{5}+3 \left (44+12 \sqrt {3}\, \sqrt {23}\right )^{\frac {1}{3}}-\frac {33}{5}\right ) \cos \left (\frac {\sqrt {3}\, \left (\left (44+12 \sqrt {3}\, \sqrt {23}\right )^{\frac {2}{3}}+20\right ) \ln \left (x \right )}{12 \left (44+12 \sqrt {3}\, \sqrt {23}\right )^{\frac {1}{3}}}\right )+\sin \left (\frac {\sqrt {3}\, \left (\left (44+12 \sqrt {3}\, \sqrt {23}\right )^{\frac {2}{3}}+20\right ) \ln \left (x \right )}{12 \left (44+12 \sqrt {3}\, \sqrt {23}\right )^{\frac {1}{3}}}\right ) \left (\sqrt {3}\, \left (44+12 \sqrt {3}\, \sqrt {23}\right )^{\frac {1}{3}}+\frac {11 \sqrt {3}}{5}+\frac {9 \sqrt {23}}{5}\right )\right ) \sqrt {23}}{69 \left (44+12 \sqrt {3}\, \sqrt {23}\right )^{\frac {2}{3}} x^{2}}\right ) \, dx}\\ &= \text {Expression too large to display} \end {align*}

\begin {align*} U_3 &= (-1)^{3-3} \int { \frac {F(x) W_3(x) }{a W(x)} \, dx}\\ &= (-1)^{0} \int { \frac { \left (2 x^{3}-\ln \left (x \right )\right ) \left (\frac {\sqrt {3}\, \left (\left (44+12 \sqrt {3}\, \sqrt {23}\right )^{\frac {2}{3}}+20\right ) x^{\frac {\left (44+12 \sqrt {3}\, \sqrt {23}\right )^{\frac {2}{3}}+2 \left (44+12 \sqrt {3}\, \sqrt {23}\right )^{\frac {1}{3}}-20}{6 \left (44+12 \sqrt {3}\, \sqrt {23}\right )^{\frac {1}{3}}}}}{12 \left (44+12 \sqrt {3}\, \sqrt {23}\right )^{\frac {1}{3}}}\right )}{\left (x^{3}\right ) \left (\frac {\sqrt {23}}{2 x}\right )} \, dx} \\ &= \int { \frac {\frac {\left (2 x^{3}-\ln \left (x \right )\right ) \sqrt {3}\, \left (\left (44+12 \sqrt {3}\, \sqrt {23}\right )^{\frac {2}{3}}+20\right ) x^{\frac {\left (44+12 \sqrt {3}\, \sqrt {23}\right )^{\frac {2}{3}}+2 \left (44+12 \sqrt {3}\, \sqrt {23}\right )^{\frac {1}{3}}-20}{6 \left (44+12 \sqrt {3}\, \sqrt {23}\right )^{\frac {1}{3}}}}}{12 \left (44+12 \sqrt {3}\, \sqrt {23}\right )^{\frac {1}{3}}}}{\frac {x^{2} \sqrt {23}}{2}} \, dx}\\ &= \int {\left (-\frac {x^{\frac {\left (44+12 \sqrt {69}\right )^{\frac {2}{3}}-10 \left (44+12 \sqrt {69}\right )^{\frac {1}{3}}-20}{6 \left (44+12 \sqrt {69}\right )^{\frac {1}{3}}}} \left (-2 x^{3}+\ln \left (x \right )\right ) \left (\left (44+12 \sqrt {69}\right )^{\frac {2}{3}}+20\right ) \sqrt {69}}{138 \left (44+12 \sqrt {69}\right )^{\frac {1}{3}}}\right ) \, dx} \\ &= \left (1+\frac {37 \left (44+12 \sqrt {69}\right )^{\frac {1}{3}} \sqrt {69}}{1380}-\frac {\left (44+12 \sqrt {69}\right )^{\frac {1}{3}}}{20}-\frac {\sqrt {69}\, \left (44+12 \sqrt {69}\right )^{\frac {2}{3}}}{690}+\frac {\left (44+12 \sqrt {69}\right )^{\frac {2}{3}}}{20}\right ) x \,{\mathrm e}^{\frac {\left (\left (44+12 \sqrt {69}\right )^{\frac {2}{3}}-10 \left (44+12 \sqrt {69}\right )^{\frac {1}{3}}-20\right ) \ln \left (x \right )}{6 \left (44+12 \sqrt {69}\right )^{\frac {1}{3}}}}+\left (\frac {2}{51}+\frac {19 \left (44+12 \sqrt {69}\right )^{\frac {1}{3}} \sqrt {69}}{3910}+\frac {7 \left (44+12 \sqrt {69}\right )^{\frac {1}{3}}}{510}-\frac {37 \sqrt {69}\, \left (44+12 \sqrt {69}\right )^{\frac {2}{3}}}{39100}+\frac {59 \left (44+12 \sqrt {69}\right )^{\frac {2}{3}}}{5100}\right ) x^{4} {\mathrm e}^{\frac {\left (\left (44+12 \sqrt {69}\right )^{\frac {2}{3}}-10 \left (44+12 \sqrt {69}\right )^{\frac {1}{3}}-20\right ) \ln \left (x \right )}{6 \left (44+12 \sqrt {69}\right )^{\frac {1}{3}}}}+\left (-\frac {\left (44+12 \sqrt {69}\right )^{\frac {1}{3}}}{30}-\frac {\sqrt {69}\, \left (44+12 \sqrt {69}\right )^{\frac {2}{3}}}{2300}+\frac {7 \left (44+12 \sqrt {69}\right )^{\frac {2}{3}}}{300}+\frac {1}{3}+\frac {3 \left (44+12 \sqrt {69}\right )^{\frac {1}{3}} \sqrt {69}}{230}\right ) x \ln \left (x \right ) {\mathrm e}^{\frac {\left (\left (44+12 \sqrt {69}\right )^{\frac {2}{3}}-10 \left (44+12 \sqrt {69}\right )^{\frac {1}{3}}-20\right ) \ln \left (x \right )}{6 \left (44+12 \sqrt {69}\right )^{\frac {1}{3}}}} \\ &= \left (1+\frac {37 \left (44+12 \sqrt {69}\right )^{\frac {1}{3}} \sqrt {69}}{1380}-\frac {\left (44+12 \sqrt {69}\right )^{\frac {1}{3}}}{20}-\frac {\sqrt {69}\, \left (44+12 \sqrt {69}\right )^{\frac {2}{3}}}{690}+\frac {\left (44+12 \sqrt {69}\right )^{\frac {2}{3}}}{20}\right ) x \,{\mathrm e}^{\frac {\left (\left (44+12 \sqrt {69}\right )^{\frac {2}{3}}-10 \left (44+12 \sqrt {69}\right )^{\frac {1}{3}}-20\right ) \ln \left (x \right )}{6 \left (44+12 \sqrt {69}\right )^{\frac {1}{3}}}}+\left (\frac {2}{51}+\frac {19 \left (44+12 \sqrt {69}\right )^{\frac {1}{3}} \sqrt {69}}{3910}+\frac {7 \left (44+12 \sqrt {69}\right )^{\frac {1}{3}}}{510}-\frac {37 \sqrt {69}\, \left (44+12 \sqrt {69}\right )^{\frac {2}{3}}}{39100}+\frac {59 \left (44+12 \sqrt {69}\right )^{\frac {2}{3}}}{5100}\right ) x^{4} {\mathrm e}^{\frac {\left (\left (44+12 \sqrt {69}\right )^{\frac {2}{3}}-10 \left (44+12 \sqrt {69}\right )^{\frac {1}{3}}-20\right ) \ln \left (x \right )}{6 \left (44+12 \sqrt {69}\right )^{\frac {1}{3}}}}+\left (-\frac {\left (44+12 \sqrt {69}\right )^{\frac {1}{3}}}{30}-\frac {\sqrt {69}\, \left (44+12 \sqrt {69}\right )^{\frac {2}{3}}}{2300}+\frac {7 \left (44+12 \sqrt {69}\right )^{\frac {2}{3}}}{300}+\frac {1}{3}+\frac {3 \left (44+12 \sqrt {69}\right )^{\frac {1}{3}} \sqrt {69}}{230}\right ) x \ln \left (x \right ) {\mathrm e}^{\frac {\left (\left (44+12 \sqrt {69}\right )^{\frac {2}{3}}-10 \left (44+12 \sqrt {69}\right )^{\frac {1}{3}}-20\right ) \ln \left (x \right )}{6 \left (44+12 \sqrt {69}\right )^{\frac {1}{3}}}} \end {align*}

Now that all the $U_i$ functions have been determined, the particular solution is found from \[ y_p = U_1 y_1+U_2 y_2+U_3 y_3 \] Hence \begin {equation*} \begin {split} y_p &= \left (\text {Expression too large to display}\right ) \left (x^{\frac {\left (44+12 \sqrt {69}\right )^{\frac {1}{3}}}{12}-\frac {5}{3 \left (44+12 \sqrt {69}\right )^{\frac {1}{3}}}+\frac {2}{3}} \cos \left (\frac {\sqrt {3}\, \left (-\frac {\left (44+12 \sqrt {69}\right )^{\frac {1}{3}}}{6}-\frac {10}{3 \left (44+12 \sqrt {69}\right )^{\frac {1}{3}}}\right ) \ln \left (x \right )}{2}\right )\right ) \\ &+\left (\text {Expression too large to display}\right ) \left (-x^{\frac {\left (44+12 \sqrt {69}\right )^{\frac {1}{3}}}{12}-\frac {5}{3 \left (44+12 \sqrt {69}\right )^{\frac {1}{3}}}+\frac {2}{3}} \sin \left (\frac {\sqrt {3}\, \left (-\frac {\left (44+12 \sqrt {69}\right )^{\frac {1}{3}}}{6}-\frac {10}{3 \left (44+12 \sqrt {69}\right )^{\frac {1}{3}}}\right ) \ln \left (x \right )}{2}\right )\right ) \\ &+\left (\left (1+\frac {37 \left (44+12 \sqrt {69}\right )^{\frac {1}{3}} \sqrt {69}}{1380}-\frac {\left (44+12 \sqrt {69}\right )^{\frac {1}{3}}}{20}-\frac {\sqrt {69}\, \left (44+12 \sqrt {69}\right )^{\frac {2}{3}}}{690}+\frac {\left (44+12 \sqrt {69}\right )^{\frac {2}{3}}}{20}\right ) x \,{\mathrm e}^{\frac {\left (\left (44+12 \sqrt {69}\right )^{\frac {2}{3}}-10 \left (44+12 \sqrt {69}\right )^{\frac {1}{3}}-20\right ) \ln \left (x \right )}{6 \left (44+12 \sqrt {69}\right )^{\frac {1}{3}}}}+\left (\frac {2}{51}+\frac {19 \left (44+12 \sqrt {69}\right )^{\frac {1}{3}} \sqrt {69}}{3910}+\frac {7 \left (44+12 \sqrt {69}\right )^{\frac {1}{3}}}{510}-\frac {37 \sqrt {69}\, \left (44+12 \sqrt {69}\right )^{\frac {2}{3}}}{39100}+\frac {59 \left (44+12 \sqrt {69}\right )^{\frac {2}{3}}}{5100}\right ) x^{4} {\mathrm e}^{\frac {\left (\left (44+12 \sqrt {69}\right )^{\frac {2}{3}}-10 \left (44+12 \sqrt {69}\right )^{\frac {1}{3}}-20\right ) \ln \left (x \right )}{6 \left (44+12 \sqrt {69}\right )^{\frac {1}{3}}}}+\left (-\frac {\left (44+12 \sqrt {69}\right )^{\frac {1}{3}}}{30}-\frac {\sqrt {69}\, \left (44+12 \sqrt {69}\right )^{\frac {2}{3}}}{2300}+\frac {7 \left (44+12 \sqrt {69}\right )^{\frac {2}{3}}}{300}+\frac {1}{3}+\frac {3 \left (44+12 \sqrt {69}\right )^{\frac {1}{3}} \sqrt {69}}{230}\right ) x \ln \left (x \right ) {\mathrm e}^{\frac {\left (\left (44+12 \sqrt {69}\right )^{\frac {2}{3}}-10 \left (44+12 \sqrt {69}\right )^{\frac {1}{3}}-20\right ) \ln \left (x \right )}{6 \left (44+12 \sqrt {69}\right )^{\frac {1}{3}}}}\right ) \left (x^{-\frac {\left (44+12 \sqrt {69}\right )^{\frac {1}{3}}}{6}+\frac {10}{3 \left (44+12 \sqrt {69}\right )^{\frac {1}{3}}}+\frac {2}{3}}\right ) \end {split} \end {equation*} Therefore the particular solution is \[ y_p = \text {Expression too large to display} \] Therefore the general solution is \begin{align*} y &= y_h + y_p \\ &= \left (x^{\frac {\left (44+12 \sqrt {69}\right )^{\frac {1}{3}}}{12}-\frac {5}{3 \left (44+12 \sqrt {69}\right )^{\frac {1}{3}}}+\frac {2}{3}} \left (c_{1} \cos \left (\frac {\sqrt {3}\, \left (-\frac {\left (44+12 \sqrt {69}\right )^{\frac {1}{3}}}{6}-\frac {10}{3 \left (44+12 \sqrt {69}\right )^{\frac {1}{3}}}\right ) \ln \left (x \right )}{2}\right )-c_{2} \sin \left (\frac {\sqrt {3}\, \left (-\frac {\left (44+12 \sqrt {69}\right )^{\frac {1}{3}}}{6}-\frac {10}{3 \left (44+12 \sqrt {69}\right )^{\frac {1}{3}}}\right ) \ln \left (x \right )}{2}\right )\right )+c_{3} x^{-\frac {\left (44+12 \sqrt {69}\right )^{\frac {1}{3}}}{6}+\frac {10}{3 \left (44+12 \sqrt {69}\right )^{\frac {1}{3}}}+\frac {2}{3}}\right ) + \left (\text {Expression too large to display}\right ) \\ \end{align*}

Summary

The solution(s) found are the following \begin{align*} \tag{1} \text {Expression too large to display} \\ \end{align*}

Veriﬁcation of solutions

\[ \text {Expression too large to display} \] Warning, solution could not be veriﬁed

4.67.1 Maple step by step solution

\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & y^{\prime \prime \prime } x^{3}+x^{2} y^{\prime \prime }+2 y^{\prime } x -y=2 x^{3}-\ln \left (x \right ) \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 3 \\ {} & {} & y^{\prime \prime \prime } \end {array} \]

Maple trace

`Methods for third order ODEs: 
--- Trying classification methods --- 
trying a quadrature 
trying high order exact linear fully integrable 
trying differential order: 3; linear nonhomogeneous with symmetry [0,1] 
trying high order linear exact nonhomogeneous 
trying differential order: 3; missing the dependent variable 
checking if the LODE is of Euler type 
<- LODE of Euler type successful 
Euler equation successful`

\[ y(x)\to \frac {i \left (\text {Root}\left [\text {$\#$1}^3-2 \text {$\#$1}^2+3 \text {$\#$1}-1\&,1\right ]-\text {Root}\left [\text {$\#$1}^3-2 \text {$\#$1}^2+3 \text {$\#$1}-1\&,2\right ]\right ) \left (\frac {2 x^3}{3-\text {Root}\left [\text {$\#$1}^3-2 \text {$\#$1}^2+3 \text {$\#$1}-1\&,3\right ]}+\frac {\log (x)}{\text {Root}\left [\text {$\#$1}^3-2 \text {$\#$1}^2+3 \text {$\#$1}-1\&,3\right ]}+\frac {1}{\text {Root}\left [\text {$\#$1}^3-2 \text {$\#$1}^2+3 \text {$\#$1}-1\&,3\right ]^2}\right )}{\sqrt {23}}-\frac {i \left (\text {Root}\left [\text {$\#$1}^3-2 \text {$\#$1}^2+3 \text {$\#$1}-1\&,1\right ]-\text {Root}\left [\text {$\#$1}^3-2 \text {$\#$1}^2+3 \text {$\#$1}-1\&,3\right ]\right ) \left (\frac {2 x^3}{3-\text {Root}\left [\text {$\#$1}^3-2 \text {$\#$1}^2+3 \text {$\#$1}-1\&,2\right ]}+\frac {\log (x)}{\text {Root}\left [\text {$\#$1}^3-2 \text {$\#$1}^2+3 \text {$\#$1}-1\&,2\right ]}+\frac {1}{\text {Root}\left [\text {$\#$1}^3-2 \text {$\#$1}^2+3 \text {$\#$1}-1\&,2\right ]^2}\right )}{\sqrt {23}}+\frac {i \left (\text {Root}\left [\text {$\#$1}^3-2 \text {$\#$1}^2+3 \text {$\#$1}-1\&,2\right ]-\text {Root}\left [\text {$\#$1}^3-2 \text {$\#$1}^2+3 \text {$\#$1}-1\&,3\right ]\right ) x^{\text {Root}\left [\text {$\#$1}^3-2 \text {$\#$1}^2+3 \text {$\#$1}-1\&,1\right ]+\text {Root}\left [\text {$\#$1}^3-2 \text {$\#$1}^2+3 \text {$\#$1}-1\&,2\right ]+\text {Root}\left [\text {$\#$1}^3-2 \text {$\#$1}^2+3 \text {$\#$1}-1\&,3\right ]-2} \left (\frac {2 x^3}{\text {Root}\left [\text {$\#$1}^3-2 \text {$\#$1}^2+3 \text {$\#$1}-1\&,2\right ]+\text {Root}\left [\text {$\#$1}^3-2 \text {$\#$1}^2+3 \text {$\#$1}-1\&,3\right ]+1}-\frac {\log (x)}{\text {Root}\left [\text {$\#$1}^3-2 \text {$\#$1}^2+3 \text {$\#$1}-1\&,2\right ]+\text {Root}\left [\text {$\#$1}^3-2 \text {$\#$1}^2+3 \text {$\#$1}-1\&,3\right ]-2}+\frac {1}{\left (\text {Root}\left [\text {$\#$1}^3-2 \text {$\#$1}^2+3 \text {$\#$1}-1\&,2\right ]+\text {Root}\left [\text {$\#$1}^3-2 \text {$\#$1}^2+3 \text {$\#$1}-1\&,3\right ]-2\right )^2}\right )}{\sqrt {23}}+c_1 x^{\text {Root}\left [\text {$\#$1}^3-2 \text {$\#$1}^2+3 \text {$\#$1}-1\&,1\right ]}+c_3 x^{\text {Root}\left [\text {$\#$1}^3-2 \text {$\#$1}^2+3 \text {$\#$1}-1\&,3\right ]}+c_2 x^{\text {Root}\left [\text {$\#$1}^3-2 \text {$\#$1}^2+3 \text {$\#$1}-1\&,2\right ]} \]

4.67 problem 1517

4.67.1 Maple step by step solution