Internal problem ID [9629]
Internal file name [OUTPUT/8571_Monday_June_06_2022_04_12_54_AM_57471240/index.tex]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 2, linear second order
Problem number: 1302.
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, _with_linear_symmetries]]
Unable to solve or complete the solution.
\[ \boxed {\operatorname {A2} \left (a x +b \right )^{2} y^{\prime \prime }+\operatorname {A1} \left (a x +b \right ) y^{\prime }+\operatorname {A0} \left (a x +b \right ) y=0} \]
Maple trace
`Methods for second order ODEs: --- Trying classification methods --- trying a quadrature 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 <- hypergeometric successful <- special function solution successful`
✓ Solution by Maple
Time used: 0.094 (sec). Leaf size: 98
dsolve(A2*(a*x+b)^2*diff(diff(y(x),x),x)+A1*(a*x+b)*diff(y(x),x)+A0*(a*x+b)*y(x)=0,y(x), singsol=all)
\[ y \left (x \right ) = \left (a x +b \right )^{-\frac {-a \operatorname {A2} +\operatorname {A1}}{2 a \operatorname {A2}}} \left (\operatorname {BesselY}\left (\frac {a \operatorname {A2} -\operatorname {A1}}{a \operatorname {A2}}, 2 \sqrt {\operatorname {A0}}\, \sqrt {\frac {a x +b}{a^{2} \operatorname {A2}}}\right ) c_{2} +\operatorname {BesselJ}\left (\frac {a \operatorname {A2} -\operatorname {A1}}{a \operatorname {A2}}, 2 \sqrt {\operatorname {A0}}\, \sqrt {\frac {a x +b}{a^{2} \operatorname {A2}}}\right ) c_{1} \right ) \]
✓ Solution by Mathematica
Time used: 0.172 (sec). Leaf size: 165
DSolve[A0*(b + a*x)*y[x] + A1*(b + a*x)*y'[x] + A2*(b + a*x)^2*y''[x] == 0,y[x],x,IncludeSingularSolutions -> True]
\[ y(x)\to (-1)^{-\frac {\text {A1}}{a \text {A2}}} \left (\frac {b}{a}+x\right )^{\frac {\text {A1}}{2 a \text {A2}}} (\text {A2} (a x+b))^{-\frac {\text {A1}}{2 a \text {A2}}} \left (-\frac {\text {A0} (a x+b)}{a^2 \text {A2}}\right )^{\frac {1}{2}-\frac {\text {A1}}{2 a \text {A2}}} \left (c_1 (-1)^{\frac {\text {A1}}{a \text {A2}}} \operatorname {BesselI}\left (\frac {\text {A1}}{a \text {A2}}-1,2 \sqrt {-\frac {\text {A0} (b+a x)}{a^2 \text {A2}}}\right )-c_2 K_{\frac {\text {A1}}{a \text {A2}}-1}\left (2 \sqrt {-\frac {\text {A0} (b+a x)}{a^2 \text {A2}}}\right )\right ) \]