Internal problem ID [7170]
Internal file name [OUTPUT/6156_Sunday_June_05_2022_04_25_47_PM_37607482/index.tex]
Book: Own collection of miscellaneous problems
Section: section 2.0
Problem number: 33.
ODE order: 2.
ODE degree: 1.
The type(s) of ODE detected by this program : "second_order_airy", "second_order_bessel_ode"
Maple gives the following as the ode type
[[_2nd_order, _linear, _nonhomogeneous]]
\[ \boxed {y^{\prime \prime }-y x=x^{6}+x^{3}-42} \]
This is Airy ODE. It has the general form \[ a y^{\prime \prime } + b y^{\prime } + c y x = F(x) \] Where in this case \begin {align*} a &= 1\\ b &= 0\\ c &= -1\\ F &= x^{6}+x^{3}-42 \end {align*}
Therefore the solution to the homogeneous Airy ODE becomes \[ y = {\mathrm e}^{-\frac {b x}{2 a}} \left (c_{1} \operatorname {AiryAi}\left (\frac {\left (-\frac {c}{a}\right )^{\frac {1}{3}} \left (4 c x a +b^{2}\right )}{4 c a}\right )+c_{2} \operatorname {AiryBi}\left (\frac {\left (-\frac {c}{a}\right )^{\frac {1}{3}} \left (4 c x a +b^{2}\right )}{4 c a}\right )\right ) \] Substituting the values for \(a,b,c\) gives \[ y = c_{1} \operatorname {AiryAi}\left (x \right )+c_{2} \operatorname {AiryBi}\left (x \right ) \] Since this is inhomogeneous Airy ODE, then we need to find the particular solution and add that to the homogeneous above. The particular solution is found using variation of parameters. Let \begin{equation} \tag{1} y_p(x) = u_1 y_1 + u_2 y_2 \end{equation} Where \(u_1,u_2\) to be determined, and \(y_1,y_2\) are the two basis solutions (the two linearly independent solutions of the homogeneous ODE) found earlier when solving the homogeneous ODE as \begin{align*} y_1 &= \operatorname {AiryAi}\left (x \right ) \\ y_2 &= \operatorname {AiryBi}\left (x \right ) \\ \end{align*} In the Variation of parameters \(u_1,u_2\) are found using \begin{align*} \tag{2} u_1 &= -\int \frac {y_2 f(x)}{a W(x)} \\ \tag{3} u_2 &= \int \frac {y_1 f(x)}{a W(x)} \\ \end{align*} Where \(W(x)\) is the Wronskian and \(a\) is the coefficient in front of \(y''\) in the given ODE. The Wronskian is given by \(W= \begin {vmatrix} y_1 & y_{2} \\ y_{1}^{\prime } & y_{2}^{\prime } \end {vmatrix} \). Hence \[ W = \begin {vmatrix} \operatorname {AiryAi}\left (x \right ) & \operatorname {AiryBi}\left (x \right ) \\ \frac {d}{dx}\left (\operatorname {AiryAi}\left (x \right )\right ) & \frac {d}{dx}\left (\operatorname {AiryBi}\left (x \right )\right ) \end {vmatrix} \] Which gives \[ W = \begin {vmatrix} \operatorname {AiryAi}\left (x \right ) & \operatorname {AiryBi}\left (x \right ) \\ \operatorname {AiryAi}\left (1, x\right ) & \operatorname {AiryBi}\left (1, x\right ) \end {vmatrix} \] Therefore \[ W = \left (\operatorname {AiryAi}\left (x \right )\right )\left (\operatorname {AiryBi}\left (1, x\right )\right ) - \left (\operatorname {AiryBi}\left (x \right )\right )\left (\operatorname {AiryAi}\left (1, x\right )\right ) \] Which simplifies to \[ W = \operatorname {AiryAi}\left (x \right ) \operatorname {AiryBi}\left (1, x\right )-\operatorname {AiryBi}\left (x \right ) \operatorname {AiryAi}\left (1, x\right ) \] Which simplifies to \[ W = \frac {1}{\pi } \] Therefore Eq. (2) becomes \[ u_1 = -\int \frac {\operatorname {AiryBi}\left (x \right ) \left (x^{6}+x^{3}-42\right )}{\frac {1}{\pi }}\,dx \] Which simplifies to \[ u_1 = - \int \operatorname {AiryBi}\left (x \right ) \left (x^{6}+x^{3}-42\right ) \pi d x \] Hence \[ u_1 = -\frac {x \left (3^{\frac {2}{3}} \operatorname {hypergeom}\left (\left [\frac {8}{3}\right ], \left [\frac {4}{3}, \frac {11}{3}\right ], \frac {x^{3}}{9}\right ) \Gamma \left (\frac {2}{3}\right )^{2} x^{7}+\frac {16 \pi 3^{\frac {5}{6}} \operatorname {hypergeom}\left (\left [\frac {7}{3}\right ], \left [\frac {2}{3}, \frac {10}{3}\right ], \frac {x^{3}}{9}\right ) x^{6}}{21}+\frac {8 \,3^{\frac {2}{3}} \operatorname {hypergeom}\left (\left [\frac {5}{3}\right ], \left [\frac {4}{3}, \frac {8}{3}\right ], \frac {x^{3}}{9}\right ) \Gamma \left (\frac {2}{3}\right )^{2} x^{4}}{5}+\frac {4 \pi \operatorname {hypergeom}\left (\left [\frac {4}{3}\right ], \left [\frac {2}{3}, \frac {7}{3}\right ], \frac {x^{3}}{9}\right ) 3^{\frac {5}{6}} x^{3}}{3}-168 \,3^{\frac {2}{3}} \operatorname {hypergeom}\left (\left [\frac {2}{3}\right ], \left [\frac {4}{3}, \frac {5}{3}\right ], \frac {x^{3}}{9}\right ) \Gamma \left (\frac {2}{3}\right )^{2} x -224 \pi \operatorname {hypergeom}\left (\left [\frac {1}{3}\right ], \left [\frac {2}{3}, \frac {4}{3}\right ], \frac {x^{3}}{9}\right ) 3^{\frac {5}{6}}\right )}{16 \Gamma \left (\frac {2}{3}\right )} \] And Eq. (3) becomes \[ u_2 = \int \frac {\operatorname {AiryAi}\left (x \right ) \left (x^{6}+x^{3}-42\right )}{\frac {1}{\pi }}\,dx \] Which simplifies to \[ u_2 = \int \operatorname {AiryAi}\left (x \right ) \left (x^{6}+x^{3}-42\right ) \pi d x \] Hence \[ u_2 = -\frac {x \left (3^{\frac {1}{6}} \operatorname {hypergeom}\left (\left [\frac {8}{3}\right ], \left [\frac {4}{3}, \frac {11}{3}\right ], \frac {x^{3}}{9}\right ) \Gamma \left (\frac {2}{3}\right )^{2} x^{7}-\frac {16 \pi 3^{\frac {1}{3}} \operatorname {hypergeom}\left (\left [\frac {7}{3}\right ], \left [\frac {2}{3}, \frac {10}{3}\right ], \frac {x^{3}}{9}\right ) x^{6}}{21}+\frac {8 \operatorname {hypergeom}\left (\left [\frac {5}{3}\right ], \left [\frac {4}{3}, \frac {8}{3}\right ], \frac {x^{3}}{9}\right ) 3^{\frac {1}{6}} \Gamma \left (\frac {2}{3}\right )^{2} x^{4}}{5}-\frac {4 \pi 3^{\frac {1}{3}} \operatorname {hypergeom}\left (\left [\frac {4}{3}\right ], \left [\frac {2}{3}, \frac {7}{3}\right ], \frac {x^{3}}{9}\right ) x^{3}}{3}-168 \,3^{\frac {1}{6}} \Gamma \left (\frac {2}{3}\right )^{2} x \operatorname {hypergeom}\left (\left [\frac {2}{3}\right ], \left [\frac {4}{3}, \frac {5}{3}\right ], \frac {x^{3}}{9}\right )+224 \,3^{\frac {1}{3}} \operatorname {hypergeom}\left (\left [\frac {1}{3}\right ], \left [\frac {2}{3}, \frac {4}{3}\right ], \frac {x^{3}}{9}\right ) \pi \right )}{16 \Gamma \left (\frac {2}{3}\right )} \] Therefore the particular solution, from equation (1) is \[ y_p(x) = -\frac {x \left (3^{\frac {2}{3}} \operatorname {hypergeom}\left (\left [\frac {8}{3}\right ], \left [\frac {4}{3}, \frac {11}{3}\right ], \frac {x^{3}}{9}\right ) \Gamma \left (\frac {2}{3}\right )^{2} x^{7}+\frac {16 \pi 3^{\frac {5}{6}} \operatorname {hypergeom}\left (\left [\frac {7}{3}\right ], \left [\frac {2}{3}, \frac {10}{3}\right ], \frac {x^{3}}{9}\right ) x^{6}}{21}+\frac {8 \,3^{\frac {2}{3}} \operatorname {hypergeom}\left (\left [\frac {5}{3}\right ], \left [\frac {4}{3}, \frac {8}{3}\right ], \frac {x^{3}}{9}\right ) \Gamma \left (\frac {2}{3}\right )^{2} x^{4}}{5}+\frac {4 \pi \operatorname {hypergeom}\left (\left [\frac {4}{3}\right ], \left [\frac {2}{3}, \frac {7}{3}\right ], \frac {x^{3}}{9}\right ) 3^{\frac {5}{6}} x^{3}}{3}-168 \,3^{\frac {2}{3}} \operatorname {hypergeom}\left (\left [\frac {2}{3}\right ], \left [\frac {4}{3}, \frac {5}{3}\right ], \frac {x^{3}}{9}\right ) \Gamma \left (\frac {2}{3}\right )^{2} x -224 \pi \operatorname {hypergeom}\left (\left [\frac {1}{3}\right ], \left [\frac {2}{3}, \frac {4}{3}\right ], \frac {x^{3}}{9}\right ) 3^{\frac {5}{6}}\right ) \operatorname {AiryAi}\left (x \right )}{16 \Gamma \left (\frac {2}{3}\right )}-\frac {x \left (3^{\frac {1}{6}} \operatorname {hypergeom}\left (\left [\frac {8}{3}\right ], \left [\frac {4}{3}, \frac {11}{3}\right ], \frac {x^{3}}{9}\right ) \Gamma \left (\frac {2}{3}\right )^{2} x^{7}-\frac {16 \pi 3^{\frac {1}{3}} \operatorname {hypergeom}\left (\left [\frac {7}{3}\right ], \left [\frac {2}{3}, \frac {10}{3}\right ], \frac {x^{3}}{9}\right ) x^{6}}{21}+\frac {8 \operatorname {hypergeom}\left (\left [\frac {5}{3}\right ], \left [\frac {4}{3}, \frac {8}{3}\right ], \frac {x^{3}}{9}\right ) 3^{\frac {1}{6}} \Gamma \left (\frac {2}{3}\right )^{2} x^{4}}{5}-\frac {4 \pi 3^{\frac {1}{3}} \operatorname {hypergeom}\left (\left [\frac {4}{3}\right ], \left [\frac {2}{3}, \frac {7}{3}\right ], \frac {x^{3}}{9}\right ) x^{3}}{3}-168 \,3^{\frac {1}{6}} \Gamma \left (\frac {2}{3}\right )^{2} x \operatorname {hypergeom}\left (\left [\frac {2}{3}\right ], \left [\frac {4}{3}, \frac {5}{3}\right ], \frac {x^{3}}{9}\right )+224 \,3^{\frac {1}{3}} \operatorname {hypergeom}\left (\left [\frac {1}{3}\right ], \left [\frac {2}{3}, \frac {4}{3}\right ], \frac {x^{3}}{9}\right ) \pi \right ) \operatorname {AiryBi}\left (x \right )}{16 \Gamma \left (\frac {2}{3}\right )} \] Which simplifies to \[ y_p(x) = -\frac {x \left (-\frac {16 x^{6} \pi \left (\operatorname {AiryBi}\left (x \right ) 3^{\frac {1}{3}}-3^{\frac {5}{6}} \operatorname {AiryAi}\left (x \right )\right ) \operatorname {hypergeom}\left (\left [\frac {7}{3}\right ], \left [\frac {2}{3}, \frac {10}{3}\right ], \frac {x^{3}}{9}\right )}{21}+x^{7} \Gamma \left (\frac {2}{3}\right )^{2} \left (\operatorname {AiryAi}\left (x \right ) 3^{\frac {2}{3}}+\operatorname {AiryBi}\left (x \right ) 3^{\frac {1}{6}}\right ) \operatorname {hypergeom}\left (\left [\frac {8}{3}\right ], \left [\frac {4}{3}, \frac {11}{3}\right ], \frac {x^{3}}{9}\right )+224 \pi \left (\operatorname {AiryBi}\left (x \right ) 3^{\frac {1}{3}}-3^{\frac {5}{6}} \operatorname {AiryAi}\left (x \right )\right ) \operatorname {hypergeom}\left (\left [\frac {1}{3}\right ], \left [\frac {2}{3}, \frac {4}{3}\right ], \frac {x^{3}}{9}\right )+\frac {8 x \left (-105 \Gamma \left (\frac {2}{3}\right )^{2} \left (\operatorname {AiryAi}\left (x \right ) 3^{\frac {2}{3}}+\operatorname {AiryBi}\left (x \right ) 3^{\frac {1}{6}}\right ) \operatorname {hypergeom}\left (\left [\frac {2}{3}\right ], \left [\frac {4}{3}, \frac {5}{3}\right ], \frac {x^{3}}{9}\right )+x^{2} \left (-\frac {5 \pi \left (\operatorname {AiryBi}\left (x \right ) 3^{\frac {1}{3}}-3^{\frac {5}{6}} \operatorname {AiryAi}\left (x \right )\right ) \operatorname {hypergeom}\left (\left [\frac {4}{3}\right ], \left [\frac {2}{3}, \frac {7}{3}\right ], \frac {x^{3}}{9}\right )}{6}+x \operatorname {hypergeom}\left (\left [\frac {5}{3}\right ], \left [\frac {4}{3}, \frac {8}{3}\right ], \frac {x^{3}}{9}\right ) \Gamma \left (\frac {2}{3}\right )^{2} \left (\operatorname {AiryAi}\left (x \right ) 3^{\frac {2}{3}}+\operatorname {AiryBi}\left (x \right ) 3^{\frac {1}{6}}\right )\right )\right )}{5}\right )}{16 \Gamma \left (\frac {2}{3}\right )} \] Therefore the general solution is \begin{align*} y &= y_h + y_p \\ &= \left (c_{1} \operatorname {AiryAi}\left (x \right )+c_{2} \operatorname {AiryBi}\left (x \right )\right ) + \left (-\frac {x \left (-\frac {16 x^{6} \pi \left (\operatorname {AiryBi}\left (x \right ) 3^{\frac {1}{3}}-3^{\frac {5}{6}} \operatorname {AiryAi}\left (x \right )\right ) \operatorname {hypergeom}\left (\left [\frac {7}{3}\right ], \left [\frac {2}{3}, \frac {10}{3}\right ], \frac {x^{3}}{9}\right )}{21}+x^{7} \Gamma \left (\frac {2}{3}\right )^{2} \left (\operatorname {AiryAi}\left (x \right ) 3^{\frac {2}{3}}+\operatorname {AiryBi}\left (x \right ) 3^{\frac {1}{6}}\right ) \operatorname {hypergeom}\left (\left [\frac {8}{3}\right ], \left [\frac {4}{3}, \frac {11}{3}\right ], \frac {x^{3}}{9}\right )+224 \pi \left (\operatorname {AiryBi}\left (x \right ) 3^{\frac {1}{3}}-3^{\frac {5}{6}} \operatorname {AiryAi}\left (x \right )\right ) \operatorname {hypergeom}\left (\left [\frac {1}{3}\right ], \left [\frac {2}{3}, \frac {4}{3}\right ], \frac {x^{3}}{9}\right )+\frac {8 x \left (-105 \Gamma \left (\frac {2}{3}\right )^{2} \left (\operatorname {AiryAi}\left (x \right ) 3^{\frac {2}{3}}+\operatorname {AiryBi}\left (x \right ) 3^{\frac {1}{6}}\right ) \operatorname {hypergeom}\left (\left [\frac {2}{3}\right ], \left [\frac {4}{3}, \frac {5}{3}\right ], \frac {x^{3}}{9}\right )+x^{2} \left (-\frac {5 \pi \left (\operatorname {AiryBi}\left (x \right ) 3^{\frac {1}{3}}-3^{\frac {5}{6}} \operatorname {AiryAi}\left (x \right )\right ) \operatorname {hypergeom}\left (\left [\frac {4}{3}\right ], \left [\frac {2}{3}, \frac {7}{3}\right ], \frac {x^{3}}{9}\right )}{6}+x \operatorname {hypergeom}\left (\left [\frac {5}{3}\right ], \left [\frac {4}{3}, \frac {8}{3}\right ], \frac {x^{3}}{9}\right ) \Gamma \left (\frac {2}{3}\right )^{2} \left (\operatorname {AiryAi}\left (x \right ) 3^{\frac {2}{3}}+\operatorname {AiryBi}\left (x \right ) 3^{\frac {1}{6}}\right )\right )\right )}{5}\right )}{16 \Gamma \left (\frac {2}{3}\right )}\right ) \\ &= -\frac {x \left (-\frac {16 x^{6} \pi \left (\operatorname {AiryBi}\left (x \right ) 3^{\frac {1}{3}}-3^{\frac {5}{6}} \operatorname {AiryAi}\left (x \right )\right ) \operatorname {hypergeom}\left (\left [\frac {7}{3}\right ], \left [\frac {2}{3}, \frac {10}{3}\right ], \frac {x^{3}}{9}\right )}{21}+x^{7} \Gamma \left (\frac {2}{3}\right )^{2} \left (\operatorname {AiryAi}\left (x \right ) 3^{\frac {2}{3}}+\operatorname {AiryBi}\left (x \right ) 3^{\frac {1}{6}}\right ) \operatorname {hypergeom}\left (\left [\frac {8}{3}\right ], \left [\frac {4}{3}, \frac {11}{3}\right ], \frac {x^{3}}{9}\right )+224 \pi \left (\operatorname {AiryBi}\left (x \right ) 3^{\frac {1}{3}}-3^{\frac {5}{6}} \operatorname {AiryAi}\left (x \right )\right ) \operatorname {hypergeom}\left (\left [\frac {1}{3}\right ], \left [\frac {2}{3}, \frac {4}{3}\right ], \frac {x^{3}}{9}\right )+\frac {8 x \left (-105 \Gamma \left (\frac {2}{3}\right )^{2} \left (\operatorname {AiryAi}\left (x \right ) 3^{\frac {2}{3}}+\operatorname {AiryBi}\left (x \right ) 3^{\frac {1}{6}}\right ) \operatorname {hypergeom}\left (\left [\frac {2}{3}\right ], \left [\frac {4}{3}, \frac {5}{3}\right ], \frac {x^{3}}{9}\right )+x^{2} \left (-\frac {5 \pi \left (\operatorname {AiryBi}\left (x \right ) 3^{\frac {1}{3}}-3^{\frac {5}{6}} \operatorname {AiryAi}\left (x \right )\right ) \operatorname {hypergeom}\left (\left [\frac {4}{3}\right ], \left [\frac {2}{3}, \frac {7}{3}\right ], \frac {x^{3}}{9}\right )}{6}+x \operatorname {hypergeom}\left (\left [\frac {5}{3}\right ], \left [\frac {4}{3}, \frac {8}{3}\right ], \frac {x^{3}}{9}\right ) \Gamma \left (\frac {2}{3}\right )^{2} \left (\operatorname {AiryAi}\left (x \right ) 3^{\frac {2}{3}}+\operatorname {AiryBi}\left (x \right ) 3^{\frac {1}{6}}\right )\right )\right )}{5}\right )}{16 \Gamma \left (\frac {2}{3}\right )}+c_{1} \operatorname {AiryAi}\left (x \right )+c_{2} \operatorname {AiryBi}\left (x \right ) \\ \end{align*}
The solution(s) found are the following \begin{align*} \tag{1} y &= -\frac {x \left (-\frac {16 x^{6} \pi \left (\operatorname {AiryBi}\left (x \right ) 3^{\frac {1}{3}}-3^{\frac {5}{6}} \operatorname {AiryAi}\left (x \right )\right ) \operatorname {hypergeom}\left (\left [\frac {7}{3}\right ], \left [\frac {2}{3}, \frac {10}{3}\right ], \frac {x^{3}}{9}\right )}{21}+x^{7} \Gamma \left (\frac {2}{3}\right )^{2} \left (\operatorname {AiryAi}\left (x \right ) 3^{\frac {2}{3}}+\operatorname {AiryBi}\left (x \right ) 3^{\frac {1}{6}}\right ) \operatorname {hypergeom}\left (\left [\frac {8}{3}\right ], \left [\frac {4}{3}, \frac {11}{3}\right ], \frac {x^{3}}{9}\right )+224 \pi \left (\operatorname {AiryBi}\left (x \right ) 3^{\frac {1}{3}}-3^{\frac {5}{6}} \operatorname {AiryAi}\left (x \right )\right ) \operatorname {hypergeom}\left (\left [\frac {1}{3}\right ], \left [\frac {2}{3}, \frac {4}{3}\right ], \frac {x^{3}}{9}\right )+\frac {8 x \left (-105 \Gamma \left (\frac {2}{3}\right )^{2} \left (\operatorname {AiryAi}\left (x \right ) 3^{\frac {2}{3}}+\operatorname {AiryBi}\left (x \right ) 3^{\frac {1}{6}}\right ) \operatorname {hypergeom}\left (\left [\frac {2}{3}\right ], \left [\frac {4}{3}, \frac {5}{3}\right ], \frac {x^{3}}{9}\right )+x^{2} \left (-\frac {5 \pi \left (\operatorname {AiryBi}\left (x \right ) 3^{\frac {1}{3}}-3^{\frac {5}{6}} \operatorname {AiryAi}\left (x \right )\right ) \operatorname {hypergeom}\left (\left [\frac {4}{3}\right ], \left [\frac {2}{3}, \frac {7}{3}\right ], \frac {x^{3}}{9}\right )}{6}+x \operatorname {hypergeom}\left (\left [\frac {5}{3}\right ], \left [\frac {4}{3}, \frac {8}{3}\right ], \frac {x^{3}}{9}\right ) \Gamma \left (\frac {2}{3}\right )^{2} \left (\operatorname {AiryAi}\left (x \right ) 3^{\frac {2}{3}}+\operatorname {AiryBi}\left (x \right ) 3^{\frac {1}{6}}\right )\right )\right )}{5}\right )}{16 \Gamma \left (\frac {2}{3}\right )}+c_{1} \operatorname {AiryAi}\left (x \right )+c_{2} \operatorname {AiryBi}\left (x \right ) \\ \end{align*}
Verification of solutions
\[ y = -\frac {x \left (-\frac {16 x^{6} \pi \left (\operatorname {AiryBi}\left (x \right ) 3^{\frac {1}{3}}-3^{\frac {5}{6}} \operatorname {AiryAi}\left (x \right )\right ) \operatorname {hypergeom}\left (\left [\frac {7}{3}\right ], \left [\frac {2}{3}, \frac {10}{3}\right ], \frac {x^{3}}{9}\right )}{21}+x^{7} \Gamma \left (\frac {2}{3}\right )^{2} \left (\operatorname {AiryAi}\left (x \right ) 3^{\frac {2}{3}}+\operatorname {AiryBi}\left (x \right ) 3^{\frac {1}{6}}\right ) \operatorname {hypergeom}\left (\left [\frac {8}{3}\right ], \left [\frac {4}{3}, \frac {11}{3}\right ], \frac {x^{3}}{9}\right )+224 \pi \left (\operatorname {AiryBi}\left (x \right ) 3^{\frac {1}{3}}-3^{\frac {5}{6}} \operatorname {AiryAi}\left (x \right )\right ) \operatorname {hypergeom}\left (\left [\frac {1}{3}\right ], \left [\frac {2}{3}, \frac {4}{3}\right ], \frac {x^{3}}{9}\right )+\frac {8 x \left (-105 \Gamma \left (\frac {2}{3}\right )^{2} \left (\operatorname {AiryAi}\left (x \right ) 3^{\frac {2}{3}}+\operatorname {AiryBi}\left (x \right ) 3^{\frac {1}{6}}\right ) \operatorname {hypergeom}\left (\left [\frac {2}{3}\right ], \left [\frac {4}{3}, \frac {5}{3}\right ], \frac {x^{3}}{9}\right )+x^{2} \left (-\frac {5 \pi \left (\operatorname {AiryBi}\left (x \right ) 3^{\frac {1}{3}}-3^{\frac {5}{6}} \operatorname {AiryAi}\left (x \right )\right ) \operatorname {hypergeom}\left (\left [\frac {4}{3}\right ], \left [\frac {2}{3}, \frac {7}{3}\right ], \frac {x^{3}}{9}\right )}{6}+x \operatorname {hypergeom}\left (\left [\frac {5}{3}\right ], \left [\frac {4}{3}, \frac {8}{3}\right ], \frac {x^{3}}{9}\right ) \Gamma \left (\frac {2}{3}\right )^{2} \left (\operatorname {AiryAi}\left (x \right ) 3^{\frac {2}{3}}+\operatorname {AiryBi}\left (x \right ) 3^{\frac {1}{6}}\right )\right )\right )}{5}\right )}{16 \Gamma \left (\frac {2}{3}\right )}+c_{1} \operatorname {AiryAi}\left (x \right )+c_{2} \operatorname {AiryBi}\left (x \right ) \] Verified OK.
Writing the ode as \begin {align*} y^{\prime \prime } x^{2}-x^{3} y = x^{2} \left (x^{6}+x^{3}-42\right )\tag {1} \end {align*}
Let the solution be \begin {align*} y &= y_h + y_p \end {align*}
Where \(y_h\) is the solution to the homogeneous ODE and \(y_p\) is a particular solution to the non-homogeneous ODE. Bessel ode has the form \begin {align*} y^{\prime \prime } x^{2}+x y^{\prime }+\left (-n^{2}+x^{2}\right ) y = 0\tag {2} \end {align*}
The generalized form of Bessel ode is given by Bowman (1958) as the following \begin {align*} y^{\prime \prime } x^{2}+\left (1-2 \alpha \right ) x y^{\prime }+\left (\beta ^{2} \gamma ^{2} x^{2 \gamma }-n^{2} \gamma ^{2}+\alpha ^{2}\right ) y = 0\tag {3} \end {align*}
With the standard solution \begin {align*} y&=x^{\alpha } \left (c_{1} \operatorname {BesselJ}\left (n , \beta \,x^{\gamma }\right )+c_{2} \operatorname {BesselY}\left (n , \beta \,x^{\gamma }\right )\right )\tag {4} \end {align*}
Comparing (3) to (1) and solving for \(\alpha ,\beta ,n,\gamma \) gives \begin {align*} \alpha &= {\frac {1}{2}}\\ \beta &= \frac {2 i}{3}\\ n &= {\frac {1}{3}}\\ \gamma &= {\frac {3}{2}} \end {align*}
Substituting all the above into (4) gives the solution as \begin {align*} y = c_{1} \sqrt {x}\, \operatorname {BesselJ}\left (\frac {1}{3}, \frac {2 i x^{\frac {3}{2}}}{3}\right )+c_{2} \sqrt {x}\, \operatorname {BesselY}\left (\frac {1}{3}, \frac {2 i x^{\frac {3}{2}}}{3}\right ) \end {align*}
Therefore the homogeneous solution \(y_h\) is \[ y_h = c_{1} \sqrt {x}\, \operatorname {BesselJ}\left (\frac {1}{3}, \frac {2 i x^{\frac {3}{2}}}{3}\right )+c_{2} \sqrt {x}\, \operatorname {BesselY}\left (\frac {1}{3}, \frac {2 i x^{\frac {3}{2}}}{3}\right ) \] The particular solution \(y_p\) can be found using either the method of undetermined coefficients, or the method of variation of parameters. The method of variation of parameters will be used as it is more general and can be used when the coefficients of the ODE depend on \(x\) as well. Let \begin{equation} \tag{1} y_p(x) = u_1 y_1 + u_2 y_2 \end{equation} Where \(u_1,u_2\) to be determined, and \(y_1,y_2\) are the two basis solutions (the two linearly independent solutions of the homogeneous ODE) found earlier when solving the homogeneous ODE as \begin{align*} y_1 &= \operatorname {AiryAi}\left (x \right ) \\ y_2 &= \operatorname {AiryBi}\left (x \right ) \\ \end{align*} In the Variation of parameters \(u_1,u_2\) are found using \begin{align*} \tag{2} u_1 &= -\int \frac {y_2 f(x)}{a W(x)} \\ \tag{3} u_2 &= \int \frac {y_1 f(x)}{a W(x)} \\ \end{align*} Where \(W(x)\) is the Wronskian and \(a\) is the coefficient in front of \(y''\) in the given ODE. The Wronskian is given by \(W= \begin {vmatrix} y_1 & y_{2} \\ y_{1}^{\prime } & y_{2}^{\prime } \end {vmatrix} \). Hence \[ W = \begin {vmatrix} \operatorname {AiryAi}\left (x \right ) & \operatorname {AiryBi}\left (x \right ) \\ \frac {d}{dx}\left (\operatorname {AiryAi}\left (x \right )\right ) & \frac {d}{dx}\left (\operatorname {AiryBi}\left (x \right )\right ) \end {vmatrix} \] Which gives \[ W = \begin {vmatrix} \operatorname {AiryAi}\left (x \right ) & \operatorname {AiryBi}\left (x \right ) \\ \operatorname {AiryAi}\left (1, x\right ) & \operatorname {AiryBi}\left (1, x\right ) \end {vmatrix} \] Therefore \[ W = \left (\operatorname {AiryAi}\left (x \right )\right )\left (\operatorname {AiryBi}\left (1, x\right )\right ) - \left (\operatorname {AiryBi}\left (x \right )\right )\left (\operatorname {AiryAi}\left (1, x\right )\right ) \] Which simplifies to \[ W = \operatorname {AiryAi}\left (x \right ) \operatorname {AiryBi}\left (1, x\right )-\operatorname {AiryBi}\left (x \right ) \operatorname {AiryAi}\left (1, x\right ) \] Which simplifies to \[ W = \frac {1}{\pi } \] Therefore Eq. (2) becomes \[ u_1 = -\int \frac {\operatorname {AiryBi}\left (x \right ) x^{2} \left (x^{6}+x^{3}-42\right )}{\frac {x^{2}}{\pi }}\,dx \] Which simplifies to \[ u_1 = - \int \operatorname {AiryBi}\left (x \right ) \left (x^{6}+x^{3}-42\right ) \pi d x \] Hence \[ u_1 = -\frac {x \left (3^{\frac {2}{3}} \operatorname {hypergeom}\left (\left [\frac {8}{3}\right ], \left [\frac {4}{3}, \frac {11}{3}\right ], \frac {x^{3}}{9}\right ) \Gamma \left (\frac {2}{3}\right )^{2} x^{7}+\frac {16 \pi 3^{\frac {5}{6}} \operatorname {hypergeom}\left (\left [\frac {7}{3}\right ], \left [\frac {2}{3}, \frac {10}{3}\right ], \frac {x^{3}}{9}\right ) x^{6}}{21}+\frac {8 \,3^{\frac {2}{3}} \operatorname {hypergeom}\left (\left [\frac {5}{3}\right ], \left [\frac {4}{3}, \frac {8}{3}\right ], \frac {x^{3}}{9}\right ) \Gamma \left (\frac {2}{3}\right )^{2} x^{4}}{5}+\frac {4 \pi \operatorname {hypergeom}\left (\left [\frac {4}{3}\right ], \left [\frac {2}{3}, \frac {7}{3}\right ], \frac {x^{3}}{9}\right ) 3^{\frac {5}{6}} x^{3}}{3}-168 \,3^{\frac {2}{3}} \operatorname {hypergeom}\left (\left [\frac {2}{3}\right ], \left [\frac {4}{3}, \frac {5}{3}\right ], \frac {x^{3}}{9}\right ) \Gamma \left (\frac {2}{3}\right )^{2} x -224 \pi \operatorname {hypergeom}\left (\left [\frac {1}{3}\right ], \left [\frac {2}{3}, \frac {4}{3}\right ], \frac {x^{3}}{9}\right ) 3^{\frac {5}{6}}\right )}{16 \Gamma \left (\frac {2}{3}\right )} \] And Eq. (3) becomes \[ u_2 = \int \frac {\operatorname {AiryAi}\left (x \right ) x^{2} \left (x^{6}+x^{3}-42\right )}{\frac {x^{2}}{\pi }}\,dx \] Which simplifies to \[ u_2 = \int \operatorname {AiryAi}\left (x \right ) \left (x^{6}+x^{3}-42\right ) \pi d x \] Hence \[ u_2 = -\frac {x \left (3^{\frac {1}{6}} \operatorname {hypergeom}\left (\left [\frac {8}{3}\right ], \left [\frac {4}{3}, \frac {11}{3}\right ], \frac {x^{3}}{9}\right ) \Gamma \left (\frac {2}{3}\right )^{2} x^{7}-\frac {16 \pi 3^{\frac {1}{3}} \operatorname {hypergeom}\left (\left [\frac {7}{3}\right ], \left [\frac {2}{3}, \frac {10}{3}\right ], \frac {x^{3}}{9}\right ) x^{6}}{21}+\frac {8 \operatorname {hypergeom}\left (\left [\frac {5}{3}\right ], \left [\frac {4}{3}, \frac {8}{3}\right ], \frac {x^{3}}{9}\right ) 3^{\frac {1}{6}} \Gamma \left (\frac {2}{3}\right )^{2} x^{4}}{5}-\frac {4 \pi 3^{\frac {1}{3}} \operatorname {hypergeom}\left (\left [\frac {4}{3}\right ], \left [\frac {2}{3}, \frac {7}{3}\right ], \frac {x^{3}}{9}\right ) x^{3}}{3}-168 \,3^{\frac {1}{6}} \Gamma \left (\frac {2}{3}\right )^{2} x \operatorname {hypergeom}\left (\left [\frac {2}{3}\right ], \left [\frac {4}{3}, \frac {5}{3}\right ], \frac {x^{3}}{9}\right )+224 \,3^{\frac {1}{3}} \operatorname {hypergeom}\left (\left [\frac {1}{3}\right ], \left [\frac {2}{3}, \frac {4}{3}\right ], \frac {x^{3}}{9}\right ) \pi \right )}{16 \Gamma \left (\frac {2}{3}\right )} \] Therefore the particular solution, from equation (1) is \[ y_p(x) = -\frac {x \left (3^{\frac {2}{3}} \operatorname {hypergeom}\left (\left [\frac {8}{3}\right ], \left [\frac {4}{3}, \frac {11}{3}\right ], \frac {x^{3}}{9}\right ) \Gamma \left (\frac {2}{3}\right )^{2} x^{7}+\frac {16 \pi 3^{\frac {5}{6}} \operatorname {hypergeom}\left (\left [\frac {7}{3}\right ], \left [\frac {2}{3}, \frac {10}{3}\right ], \frac {x^{3}}{9}\right ) x^{6}}{21}+\frac {8 \,3^{\frac {2}{3}} \operatorname {hypergeom}\left (\left [\frac {5}{3}\right ], \left [\frac {4}{3}, \frac {8}{3}\right ], \frac {x^{3}}{9}\right ) \Gamma \left (\frac {2}{3}\right )^{2} x^{4}}{5}+\frac {4 \pi \operatorname {hypergeom}\left (\left [\frac {4}{3}\right ], \left [\frac {2}{3}, \frac {7}{3}\right ], \frac {x^{3}}{9}\right ) 3^{\frac {5}{6}} x^{3}}{3}-168 \,3^{\frac {2}{3}} \operatorname {hypergeom}\left (\left [\frac {2}{3}\right ], \left [\frac {4}{3}, \frac {5}{3}\right ], \frac {x^{3}}{9}\right ) \Gamma \left (\frac {2}{3}\right )^{2} x -224 \pi \operatorname {hypergeom}\left (\left [\frac {1}{3}\right ], \left [\frac {2}{3}, \frac {4}{3}\right ], \frac {x^{3}}{9}\right ) 3^{\frac {5}{6}}\right ) \operatorname {AiryAi}\left (x \right )}{16 \Gamma \left (\frac {2}{3}\right )}-\frac {x \left (3^{\frac {1}{6}} \operatorname {hypergeom}\left (\left [\frac {8}{3}\right ], \left [\frac {4}{3}, \frac {11}{3}\right ], \frac {x^{3}}{9}\right ) \Gamma \left (\frac {2}{3}\right )^{2} x^{7}-\frac {16 \pi 3^{\frac {1}{3}} \operatorname {hypergeom}\left (\left [\frac {7}{3}\right ], \left [\frac {2}{3}, \frac {10}{3}\right ], \frac {x^{3}}{9}\right ) x^{6}}{21}+\frac {8 \operatorname {hypergeom}\left (\left [\frac {5}{3}\right ], \left [\frac {4}{3}, \frac {8}{3}\right ], \frac {x^{3}}{9}\right ) 3^{\frac {1}{6}} \Gamma \left (\frac {2}{3}\right )^{2} x^{4}}{5}-\frac {4 \pi 3^{\frac {1}{3}} \operatorname {hypergeom}\left (\left [\frac {4}{3}\right ], \left [\frac {2}{3}, \frac {7}{3}\right ], \frac {x^{3}}{9}\right ) x^{3}}{3}-168 \,3^{\frac {1}{6}} \Gamma \left (\frac {2}{3}\right )^{2} x \operatorname {hypergeom}\left (\left [\frac {2}{3}\right ], \left [\frac {4}{3}, \frac {5}{3}\right ], \frac {x^{3}}{9}\right )+224 \,3^{\frac {1}{3}} \operatorname {hypergeom}\left (\left [\frac {1}{3}\right ], \left [\frac {2}{3}, \frac {4}{3}\right ], \frac {x^{3}}{9}\right ) \pi \right ) \operatorname {AiryBi}\left (x \right )}{16 \Gamma \left (\frac {2}{3}\right )} \] Which simplifies to \[ y_p(x) = -\frac {x \left (-\frac {16 x^{6} \pi \left (\operatorname {AiryBi}\left (x \right ) 3^{\frac {1}{3}}-3^{\frac {5}{6}} \operatorname {AiryAi}\left (x \right )\right ) \operatorname {hypergeom}\left (\left [\frac {7}{3}\right ], \left [\frac {2}{3}, \frac {10}{3}\right ], \frac {x^{3}}{9}\right )}{21}+x^{7} \Gamma \left (\frac {2}{3}\right )^{2} \left (\operatorname {AiryAi}\left (x \right ) 3^{\frac {2}{3}}+\operatorname {AiryBi}\left (x \right ) 3^{\frac {1}{6}}\right ) \operatorname {hypergeom}\left (\left [\frac {8}{3}\right ], \left [\frac {4}{3}, \frac {11}{3}\right ], \frac {x^{3}}{9}\right )+224 \pi \left (\operatorname {AiryBi}\left (x \right ) 3^{\frac {1}{3}}-3^{\frac {5}{6}} \operatorname {AiryAi}\left (x \right )\right ) \operatorname {hypergeom}\left (\left [\frac {1}{3}\right ], \left [\frac {2}{3}, \frac {4}{3}\right ], \frac {x^{3}}{9}\right )+\frac {8 x \left (-105 \Gamma \left (\frac {2}{3}\right )^{2} \left (\operatorname {AiryAi}\left (x \right ) 3^{\frac {2}{3}}+\operatorname {AiryBi}\left (x \right ) 3^{\frac {1}{6}}\right ) \operatorname {hypergeom}\left (\left [\frac {2}{3}\right ], \left [\frac {4}{3}, \frac {5}{3}\right ], \frac {x^{3}}{9}\right )+x^{2} \left (-\frac {5 \pi \left (\operatorname {AiryBi}\left (x \right ) 3^{\frac {1}{3}}-3^{\frac {5}{6}} \operatorname {AiryAi}\left (x \right )\right ) \operatorname {hypergeom}\left (\left [\frac {4}{3}\right ], \left [\frac {2}{3}, \frac {7}{3}\right ], \frac {x^{3}}{9}\right )}{6}+x \operatorname {hypergeom}\left (\left [\frac {5}{3}\right ], \left [\frac {4}{3}, \frac {8}{3}\right ], \frac {x^{3}}{9}\right ) \Gamma \left (\frac {2}{3}\right )^{2} \left (\operatorname {AiryAi}\left (x \right ) 3^{\frac {2}{3}}+\operatorname {AiryBi}\left (x \right ) 3^{\frac {1}{6}}\right )\right )\right )}{5}\right )}{16 \Gamma \left (\frac {2}{3}\right )} \] Therefore the general solution is \begin{align*} y &= y_h + y_p \\ &= \left (c_{1} \sqrt {x}\, \operatorname {BesselJ}\left (\frac {1}{3}, \frac {2 i x^{\frac {3}{2}}}{3}\right )+c_{2} \sqrt {x}\, \operatorname {BesselY}\left (\frac {1}{3}, \frac {2 i x^{\frac {3}{2}}}{3}\right )\right ) + \left (-\frac {x \left (-\frac {16 x^{6} \pi \left (\operatorname {AiryBi}\left (x \right ) 3^{\frac {1}{3}}-3^{\frac {5}{6}} \operatorname {AiryAi}\left (x \right )\right ) \operatorname {hypergeom}\left (\left [\frac {7}{3}\right ], \left [\frac {2}{3}, \frac {10}{3}\right ], \frac {x^{3}}{9}\right )}{21}+x^{7} \Gamma \left (\frac {2}{3}\right )^{2} \left (\operatorname {AiryAi}\left (x \right ) 3^{\frac {2}{3}}+\operatorname {AiryBi}\left (x \right ) 3^{\frac {1}{6}}\right ) \operatorname {hypergeom}\left (\left [\frac {8}{3}\right ], \left [\frac {4}{3}, \frac {11}{3}\right ], \frac {x^{3}}{9}\right )+224 \pi \left (\operatorname {AiryBi}\left (x \right ) 3^{\frac {1}{3}}-3^{\frac {5}{6}} \operatorname {AiryAi}\left (x \right )\right ) \operatorname {hypergeom}\left (\left [\frac {1}{3}\right ], \left [\frac {2}{3}, \frac {4}{3}\right ], \frac {x^{3}}{9}\right )+\frac {8 x \left (-105 \Gamma \left (\frac {2}{3}\right )^{2} \left (\operatorname {AiryAi}\left (x \right ) 3^{\frac {2}{3}}+\operatorname {AiryBi}\left (x \right ) 3^{\frac {1}{6}}\right ) \operatorname {hypergeom}\left (\left [\frac {2}{3}\right ], \left [\frac {4}{3}, \frac {5}{3}\right ], \frac {x^{3}}{9}\right )+x^{2} \left (-\frac {5 \pi \left (\operatorname {AiryBi}\left (x \right ) 3^{\frac {1}{3}}-3^{\frac {5}{6}} \operatorname {AiryAi}\left (x \right )\right ) \operatorname {hypergeom}\left (\left [\frac {4}{3}\right ], \left [\frac {2}{3}, \frac {7}{3}\right ], \frac {x^{3}}{9}\right )}{6}+x \operatorname {hypergeom}\left (\left [\frac {5}{3}\right ], \left [\frac {4}{3}, \frac {8}{3}\right ], \frac {x^{3}}{9}\right ) \Gamma \left (\frac {2}{3}\right )^{2} \left (\operatorname {AiryAi}\left (x \right ) 3^{\frac {2}{3}}+\operatorname {AiryBi}\left (x \right ) 3^{\frac {1}{6}}\right )\right )\right )}{5}\right )}{16 \Gamma \left (\frac {2}{3}\right )}\right ) \\ \end{align*}
The solution(s) found are the following \begin{align*} \tag{1} y &= c_{1} \sqrt {x}\, \operatorname {BesselJ}\left (\frac {1}{3}, \frac {2 i x^{\frac {3}{2}}}{3}\right )+c_{2} \sqrt {x}\, \operatorname {BesselY}\left (\frac {1}{3}, \frac {2 i x^{\frac {3}{2}}}{3}\right )-\frac {x \left (-\frac {16 x^{6} \pi \left (\operatorname {AiryBi}\left (x \right ) 3^{\frac {1}{3}}-3^{\frac {5}{6}} \operatorname {AiryAi}\left (x \right )\right ) \operatorname {hypergeom}\left (\left [\frac {7}{3}\right ], \left [\frac {2}{3}, \frac {10}{3}\right ], \frac {x^{3}}{9}\right )}{21}+x^{7} \Gamma \left (\frac {2}{3}\right )^{2} \left (\operatorname {AiryAi}\left (x \right ) 3^{\frac {2}{3}}+\operatorname {AiryBi}\left (x \right ) 3^{\frac {1}{6}}\right ) \operatorname {hypergeom}\left (\left [\frac {8}{3}\right ], \left [\frac {4}{3}, \frac {11}{3}\right ], \frac {x^{3}}{9}\right )+224 \pi \left (\operatorname {AiryBi}\left (x \right ) 3^{\frac {1}{3}}-3^{\frac {5}{6}} \operatorname {AiryAi}\left (x \right )\right ) \operatorname {hypergeom}\left (\left [\frac {1}{3}\right ], \left [\frac {2}{3}, \frac {4}{3}\right ], \frac {x^{3}}{9}\right )+\frac {8 x \left (-105 \Gamma \left (\frac {2}{3}\right )^{2} \left (\operatorname {AiryAi}\left (x \right ) 3^{\frac {2}{3}}+\operatorname {AiryBi}\left (x \right ) 3^{\frac {1}{6}}\right ) \operatorname {hypergeom}\left (\left [\frac {2}{3}\right ], \left [\frac {4}{3}, \frac {5}{3}\right ], \frac {x^{3}}{9}\right )+x^{2} \left (-\frac {5 \pi \left (\operatorname {AiryBi}\left (x \right ) 3^{\frac {1}{3}}-3^{\frac {5}{6}} \operatorname {AiryAi}\left (x \right )\right ) \operatorname {hypergeom}\left (\left [\frac {4}{3}\right ], \left [\frac {2}{3}, \frac {7}{3}\right ], \frac {x^{3}}{9}\right )}{6}+x \operatorname {hypergeom}\left (\left [\frac {5}{3}\right ], \left [\frac {4}{3}, \frac {8}{3}\right ], \frac {x^{3}}{9}\right ) \Gamma \left (\frac {2}{3}\right )^{2} \left (\operatorname {AiryAi}\left (x \right ) 3^{\frac {2}{3}}+\operatorname {AiryBi}\left (x \right ) 3^{\frac {1}{6}}\right )\right )\right )}{5}\right )}{16 \Gamma \left (\frac {2}{3}\right )} \\ \end{align*}
Verification of solutions
\[ y = c_{1} \sqrt {x}\, \operatorname {BesselJ}\left (\frac {1}{3}, \frac {2 i x^{\frac {3}{2}}}{3}\right )+c_{2} \sqrt {x}\, \operatorname {BesselY}\left (\frac {1}{3}, \frac {2 i x^{\frac {3}{2}}}{3}\right )-\frac {x \left (-\frac {16 x^{6} \pi \left (\operatorname {AiryBi}\left (x \right ) 3^{\frac {1}{3}}-3^{\frac {5}{6}} \operatorname {AiryAi}\left (x \right )\right ) \operatorname {hypergeom}\left (\left [\frac {7}{3}\right ], \left [\frac {2}{3}, \frac {10}{3}\right ], \frac {x^{3}}{9}\right )}{21}+x^{7} \Gamma \left (\frac {2}{3}\right )^{2} \left (\operatorname {AiryAi}\left (x \right ) 3^{\frac {2}{3}}+\operatorname {AiryBi}\left (x \right ) 3^{\frac {1}{6}}\right ) \operatorname {hypergeom}\left (\left [\frac {8}{3}\right ], \left [\frac {4}{3}, \frac {11}{3}\right ], \frac {x^{3}}{9}\right )+224 \pi \left (\operatorname {AiryBi}\left (x \right ) 3^{\frac {1}{3}}-3^{\frac {5}{6}} \operatorname {AiryAi}\left (x \right )\right ) \operatorname {hypergeom}\left (\left [\frac {1}{3}\right ], \left [\frac {2}{3}, \frac {4}{3}\right ], \frac {x^{3}}{9}\right )+\frac {8 x \left (-105 \Gamma \left (\frac {2}{3}\right )^{2} \left (\operatorname {AiryAi}\left (x \right ) 3^{\frac {2}{3}}+\operatorname {AiryBi}\left (x \right ) 3^{\frac {1}{6}}\right ) \operatorname {hypergeom}\left (\left [\frac {2}{3}\right ], \left [\frac {4}{3}, \frac {5}{3}\right ], \frac {x^{3}}{9}\right )+x^{2} \left (-\frac {5 \pi \left (\operatorname {AiryBi}\left (x \right ) 3^{\frac {1}{3}}-3^{\frac {5}{6}} \operatorname {AiryAi}\left (x \right )\right ) \operatorname {hypergeom}\left (\left [\frac {4}{3}\right ], \left [\frac {2}{3}, \frac {7}{3}\right ], \frac {x^{3}}{9}\right )}{6}+x \operatorname {hypergeom}\left (\left [\frac {5}{3}\right ], \left [\frac {4}{3}, \frac {8}{3}\right ], \frac {x^{3}}{9}\right ) \Gamma \left (\frac {2}{3}\right )^{2} \left (\operatorname {AiryAi}\left (x \right ) 3^{\frac {2}{3}}+\operatorname {AiryBi}\left (x \right ) 3^{\frac {1}{6}}\right )\right )\right )}{5}\right )}{16 \Gamma \left (\frac {2}{3}\right )} \] Verified OK.
Maple trace
`Methods for second order ODEs: --- Trying classification methods --- trying a quadrature trying high order exact linear fully integrable trying differential order: 2; linear nonhomogeneous with symmetry [0,1] trying a double symmetry of the form [xi=0, eta=F(x)] -> Try solving first the homogeneous part of the ODE checking if the LODE has constant coefficients checking if the LODE is of Euler type trying a symmetry of the form [xi=0, eta=F(x)] checking if the LODE is missing y -> Trying a Liouvillian solution using Kovacics algorithm <- No Liouvillian solutions exists -> Trying a solution in terms of special functions: -> Bessel <- Bessel successful <- special function solution successful <- solving first the homogeneous part of the ODE successful`
✓ Solution by Maple
Time used: 0.0 (sec). Leaf size: 23
dsolve(diff(y(x),x$2)-x*y(x)-x^6-x^3+42=0,y(x), singsol=all)
\[ y \left (x \right ) = \operatorname {AiryAi}\left (x \right ) c_{2} +\operatorname {AiryBi}\left (x \right ) c_{1} -x^{5}-21 x^{2} \]
✓ Solution by Mathematica
Time used: 1.142 (sec). Leaf size: 367
DSolve[y''[x]-x*y[x]-x^6-x^3+42==0,y[x],x,IncludeSingularSolutions -> True]
\[ y(x)\to -\frac {-126 \sqrt [3]{3} \pi x \operatorname {Gamma}\left (\frac {1}{3}\right ) \left (\sqrt {3} \operatorname {AiryAi}(x)-\operatorname {AiryBi}(x)\right ) \, _1F_2\left (\frac {1}{3};\frac {2}{3},\frac {4}{3};\frac {x^3}{9}\right )+\frac {\sqrt [6]{3} \pi x^8 \operatorname {Gamma}\left (\frac {2}{3}\right ) \operatorname {Gamma}\left (\frac {8}{3}\right ) \left (3 \operatorname {AiryAi}(x)+\sqrt {3} \operatorname {AiryBi}(x)\right ) \, _1F_2\left (\frac {8}{3};\frac {4}{3},\frac {11}{3};\frac {x^3}{9}\right )}{\operatorname {Gamma}\left (\frac {11}{3}\right )}+\frac {3 \sqrt [3]{3} \pi x^7 \operatorname {Gamma}\left (\frac {4}{3}\right ) \operatorname {Gamma}\left (\frac {7}{3}\right ) \left (\sqrt {3} \operatorname {AiryAi}(x)-\operatorname {AiryBi}(x)\right ) \, _1F_2\left (\frac {7}{3};\frac {2}{3},\frac {10}{3};\frac {x^3}{9}\right )}{\operatorname {Gamma}\left (\frac {10}{3}\right )}+\frac {\sqrt [6]{3} \pi x^5 \operatorname {Gamma}\left (\frac {2}{3}\right ) \operatorname {Gamma}\left (\frac {5}{3}\right ) \left (3 \operatorname {AiryAi}(x)+\sqrt {3} \operatorname {AiryBi}(x)\right ) \, _1F_2\left (\frac {5}{3};\frac {4}{3},\frac {8}{3};\frac {x^3}{9}\right )}{\operatorname {Gamma}\left (\frac {8}{3}\right )}+\frac {3 \sqrt [3]{3} \pi x^4 \operatorname {Gamma}\left (\frac {4}{3}\right )^2 \left (\sqrt {3} \operatorname {AiryAi}(x)-\operatorname {AiryBi}(x)\right ) \, _1F_2\left (\frac {4}{3};\frac {2}{3},\frac {7}{3};\frac {x^3}{9}\right )}{\operatorname {Gamma}\left (\frac {7}{3}\right )}-\frac {42 \sqrt [6]{3} \pi x^2 \operatorname {Gamma}\left (\frac {2}{3}\right )^2 \left (3 \operatorname {AiryAi}(x)+\sqrt {3} \operatorname {AiryBi}(x)\right ) \, _1F_2\left (\frac {2}{3};\frac {4}{3},\frac {5}{3};\frac {x^3}{9}\right )}{\operatorname {Gamma}\left (\frac {5}{3}\right )}-27 \operatorname {Gamma}\left (\frac {2}{3}\right ) \operatorname {Gamma}\left (\frac {4}{3}\right ) (c_1 \operatorname {AiryAi}(x)+c_2 \operatorname {AiryBi}(x))}{27 \operatorname {Gamma}\left (\frac {2}{3}\right ) \operatorname {Gamma}\left (\frac {4}{3}\right )} \]