Internal problem ID [9691]
Internal file name [OUTPUT/8633_Monday_June_06_2022_04_32_56_AM_75839603/index.tex]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 2, linear second order
Problem number: 1364.
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.
\[ \boxed {y^{\prime \prime }-\frac {\left (2 b c \,x^{c} \left (x^{2}-1\right )+2 \left (a -1\right ) x^{2}-2 a \right ) y^{\prime }}{x \left (x^{2}-1\right )}+\frac {\left (b^{2} c^{2} x^{2 c} \left (x^{2}-1\right )+b c \,x^{c +2} \left (2 a -c -1\right )-b c \,x^{c} \left (2 a -c +1\right )+x^{2} \left (a \left (a -1\right )-v \left (v +1\right )\right )-a \left (a +1\right )\right ) y}{x^{2} \left (x^{2}-1\right )}=0} \]
Maple trace
`Methods for second order ODEs: --- Trying classification methods --- 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 <- Legendre successful <- special function solution successful`
✓ Solution by Maple
Time used: 0.094 (sec). Leaf size: 25
dsolve(diff(diff(y(x),x),x) = 1/x*(2*b*c*x^c*(x^2-1)+2*(a-1)*x^2-2*a)/(x^2-1)*diff(y(x),x)-(b^2*c^2*x^(2*c)*(x^2-1)+b*c*x^(c+2)*(2*a-c-1)-b*c*x^c*(2*a-c+1)+x^2*(a*(a-1)-v*(v+1))-a*(a+1))/x^2/(x^2-1)*y(x),y(x), singsol=all)
\[ y \left (x \right ) = x^{a} {\mathrm e}^{b \,x^{c}} \left (c_{1} \operatorname {LegendreP}\left (v , x\right )+c_{2} \operatorname {LegendreQ}\left (v , x\right )\right ) \]
✓ Solution by Mathematica
Time used: 0.184 (sec). Leaf size: 35
DSolve[y''[x] == -(((-(a*(1 + a)) + ((-1 + a)*a - v*(1 + v))*x^2 - b*(1 + 2*a - c)*c*x^c + b*(-1 + 2*a - c)*c*x^(2 + c) + b^2*c^2*x^(2*c)*(-1 + x^2))*y[x])/(x^2*(-1 + x^2))) + ((-2*a + 2*(-1 + a)*x^2 + 2*b*c*x^c*(-1 + x^2))*y'[x])/(x*(-1 + x^2)),y[x],x,IncludeSingularSolutions -> True]
\[ y(x)\to \left (x^c\right )^{a/c} e^{b x^c} (c_1 \operatorname {LegendreP}(v,x)+c_2 \operatorname {LegendreQ}(v,x)) \]