Internal problem ID [9668]
Internal file name [OUTPUT/8610_Monday_June_06_2022_04_26_01_AM_45577581/index.tex]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 2, linear second order
Problem number: 1341.
ODE order: 2.
ODE degree: 1.
The type(s) of ODE detected by this program : "unknown"
Maple gives the following as the ode type
[[_2nd_order, _linear, _nonhomogeneous]]
Unable to solve or complete the solution.
\[ \boxed {y^{\prime \prime }+\frac {\left (2 x a +b \right ) y^{\prime }}{x \left (x a +b \right )}+\frac {\left (a v x -b \right ) y}{\left (x a +b \right ) x^{2}}=A x} \]
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 -> elliptic -> Legendre -> Kummer -> hyper3: Equivalence to 1F1 under a power @ Moebius -> hypergeometric -> heuristic approach <- heuristic approach successful -> solution has integrals; searching for one without integrals... -> hyper3: Equivalence to 2F1, 1F1 or 0F1 under a power @ Moebius <- hyper3 successful: received ODE is equivalent to the 2F1 ODE <- hypergeometric solution without integrals succesful <- hypergeometric successful <- special function solution successful <- solving first the homogeneous part of the ODE successful`
✓ Solution by Maple
Time used: 0.265 (sec). Leaf size: 194
dsolve(diff(diff(y(x),x),x) = -1/x*(2*a*x+b)/(a*x+b)*diff(y(x),x)-(a*v*x-b)/(a*x+b)/x^2*y(x)+A*x,y(x), singsol=all)
\[ y \left (x \right ) = \frac {x^{-\frac {\sqrt {1-4 v}}{2}} a^{2} c_{1} \left (v +6\right ) \left (2+v \right ) \left (v +12\right ) \operatorname {hypergeom}\left (\left [-\frac {1}{2}+\frac {\sqrt {1-4 v}}{2}, \frac {3}{2}+\frac {\sqrt {1-4 v}}{2}\right ], \left [1+\sqrt {1-4 v}\right ], -\frac {b}{a x}\right )-3 b^{2} A \left (v +4\right ) x^{\frac {3}{2}}+\left (2+v \right ) \left (A b \left (v +4\right ) x^{\frac {5}{2}}+\left (A \,x^{\frac {7}{2}}+\left (v +12\right ) x^{\frac {\sqrt {1-4 v}}{2}} \operatorname {hypergeom}\left (\left [-\frac {1}{2}-\frac {\sqrt {1-4 v}}{2}, \frac {3}{2}-\frac {\sqrt {1-4 v}}{2}\right ], \left [1-\sqrt {1-4 v}\right ], -\frac {b}{a x}\right ) c_{2} \right ) \left (v +6\right ) a \right ) a}{\sqrt {x}\, a^{2} \left (v +6\right ) \left (2+v \right ) \left (v +12\right )} \]
✓ Solution by Mathematica
Time used: 71.383 (sec). Leaf size: 725
DSolve[y''[x] == A*x - ((-b + a*v*x)*y[x])/(x^2*(b + a*x)) - ((b + 2*a*x)*y'[x])/(x*(b + a*x)),y[x],x,IncludeSingularSolutions -> True]
\[ y(x)\to \frac {a x \operatorname {Hypergeometric2F1}\left (\frac {1}{2} \left (3-\sqrt {1-4 v}\right ),\frac {1}{2} \left (\sqrt {1-4 v}+3\right ),3,-\frac {a x}{b}\right ) \left (\int _1^x-\frac {3 A b^2 K[1] G_{2,2}^{2,0}\left (-\frac {a K[1]}{b}| \begin {array}{c} \frac {1}{2} \left (1-\sqrt {1-4 v}\right ),\frac {1}{2} \left (\sqrt {1-4 v}+1\right ) \\ -1,1 \\ \end {array} \right )}{a \left (\left (a (v+2) \operatorname {Hypergeometric2F1}\left (\frac {1}{2} \left (5-\sqrt {1-4 v}\right ),\frac {1}{2} \left (\sqrt {1-4 v}+5\right ),4,-\frac {a K[1]}{b}\right ) K[1]-3 b \operatorname {Hypergeometric2F1}\left (\frac {1}{2} \left (3-\sqrt {1-4 v}\right ),\frac {1}{2} \left (\sqrt {1-4 v}+3\right ),3,-\frac {a K[1]}{b}\right )\right ) G_{2,2}^{2,0}\left (-\frac {a K[1]}{b}| \begin {array}{c} \frac {1}{2} \left (1-\sqrt {1-4 v}\right ),\frac {1}{2} \left (\sqrt {1-4 v}+1\right ) \\ -1,1 \\ \end {array} \right )+3 a \operatorname {Hypergeometric2F1}\left (\frac {1}{2} \left (3-\sqrt {1-4 v}\right ),\frac {1}{2} \left (\sqrt {1-4 v}+3\right ),3,-\frac {a K[1]}{b}\right ) K[1] G_{3,3}^{3,0}\left (-\frac {a K[1]}{b}| \begin {array}{c} -1,\frac {1}{2} \left (-\sqrt {1-4 v}-1\right ),\frac {1}{2} \left (\sqrt {1-4 v}-1\right ) \\ -2,0,0 \\ \end {array} \right )\right )}dK[1]+c_1\right )}{b}+G_{2,2}^{2,0}\left (-\frac {a x}{b}| \begin {array}{c} \frac {1}{2} \left (1-\sqrt {1-4 v}\right ),\frac {1}{2} \left (\sqrt {1-4 v}+1\right ) \\ -1,1 \\ \end {array} \right ) \left (\int _1^x-\frac {3 A b \operatorname {Hypergeometric2F1}\left (\frac {1}{2} \left (3-\sqrt {1-4 v}\right ),\frac {1}{2} \left (\sqrt {1-4 v}+3\right ),3,-\frac {a K[2]}{b}\right ) K[2]^2}{\left (3 b \operatorname {Hypergeometric2F1}\left (\frac {1}{2} \left (3-\sqrt {1-4 v}\right ),\frac {1}{2} \left (\sqrt {1-4 v}+3\right ),3,-\frac {a K[2]}{b}\right )-a (v+2) \operatorname {Hypergeometric2F1}\left (\frac {1}{2} \left (5-\sqrt {1-4 v}\right ),\frac {1}{2} \left (\sqrt {1-4 v}+5\right ),4,-\frac {a K[2]}{b}\right ) K[2]\right ) G_{2,2}^{2,0}\left (-\frac {a K[2]}{b}| \begin {array}{c} \frac {1}{2} \left (1-\sqrt {1-4 v}\right ),\frac {1}{2} \left (\sqrt {1-4 v}+1\right ) \\ -1,1 \\ \end {array} \right )-3 a \operatorname {Hypergeometric2F1}\left (\frac {1}{2} \left (3-\sqrt {1-4 v}\right ),\frac {1}{2} \left (\sqrt {1-4 v}+3\right ),3,-\frac {a K[2]}{b}\right ) K[2] G_{3,3}^{3,0}\left (-\frac {a K[2]}{b}| \begin {array}{c} -1,\frac {1}{2} \left (-\sqrt {1-4 v}-1\right ),\frac {1}{2} \left (\sqrt {1-4 v}-1\right ) \\ -2,0,0 \\ \end {array} \right )}dK[2]+c_2\right ) \]