Internal problem ID [9652]
Internal file name [OUTPUT/8594_Monday_June_06_2022_04_18_29_AM_85344772/index.tex]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 2, linear second order
Problem number: 1325.
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.
\[ \boxed {y^{\prime \prime }+\frac {\left (\left (a +b +1\right ) x +\alpha +\beta -1\right ) y^{\prime }}{x \left (x -1\right )}+\frac {\left (a b x -\alpha \beta \right ) y}{x^{2} \left (x -1\right )}=0} \]
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & \left (\frac {d}{d x}y^{\prime }\right ) x^{2} \left (x -1\right )+\left (\left (a +b +1\right ) x +\alpha +\beta -1\right ) x y^{\prime }+\left (a b x -\alpha \beta \right ) y=0 \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 2 \\ {} & {} & \frac {d}{d x}y^{\prime } \\ \bullet & {} & \textrm {Isolate 2nd derivative}\hspace {3pt} \\ {} & {} & \frac {d}{d x}y^{\prime }=-\frac {\left (a b x -\alpha \beta \right ) y}{x^{2} \left (x -1\right )}-\frac {\left (x a +x b +\alpha +\beta +x -1\right ) y^{\prime }}{x \left (x -1\right )} \\ \bullet & {} & \textrm {Group terms with}\hspace {3pt} y\hspace {3pt}\textrm {on the lhs of the ODE and the rest on the rhs of the ODE; ODE is linear}\hspace {3pt} \\ {} & {} & \frac {d}{d x}y^{\prime }+\frac {\left (x a +x b +\alpha +\beta +x -1\right ) y^{\prime }}{x \left (x -1\right )}+\frac {\left (a b x -\alpha \beta \right ) y}{x^{2} \left (x -1\right )}=0 \\ \square & {} & \textrm {Check to see if}\hspace {3pt} x_{0}\hspace {3pt}\textrm {is a regular singular point}\hspace {3pt} \\ {} & \circ & \textrm {Define functions}\hspace {3pt} \\ {} & {} & \left [P_{2}\left (x \right )=\frac {x a +x b +\alpha +\beta +x -1}{x \left (x -1\right )}, P_{3}\left (x \right )=\frac {a b x -\alpha \beta }{x^{2} \left (x -1\right )}\right ] \\ {} & \circ & x \cdot P_{2}\left (x \right )\textrm {is analytic at}\hspace {3pt} x =0 \\ {} & {} & \left (x \cdot P_{2}\left (x \right )\right )\bigg | {\mstack {}{_{x \hiderel {=}0}}}=-\beta -\alpha +1 \\ {} & \circ & x^{2}\cdot P_{3}\left (x \right )\textrm {is analytic at}\hspace {3pt} x =0 \\ {} & {} & \left (x^{2}\cdot P_{3}\left (x \right )\right )\bigg | {\mstack {}{_{x \hiderel {=}0}}}=\alpha \beta \\ {} & \circ & x =0\textrm {is a regular singular point}\hspace {3pt} \\ & {} & \textrm {Check to see if}\hspace {3pt} x_{0}\hspace {3pt}\textrm {is a regular singular point}\hspace {3pt} \\ {} & {} & x_{0}=0 \\ \bullet & {} & \textrm {Multiply by denominators}\hspace {3pt} \\ {} & {} & \left (\frac {d}{d x}y^{\prime }\right ) x^{2} \left (x -1\right )+\left (x a +x b +\alpha +\beta +x -1\right ) y^{\prime } x +\left (a b x -\alpha \beta \right ) y=0 \\ \bullet & {} & \textrm {Assume series solution for}\hspace {3pt} y \\ {} & {} & y=\moverset {\infty }{\munderset {k =0}{\sum }}a_{k} x^{k +r} \\ \square & {} & \textrm {Rewrite ODE with series expansions}\hspace {3pt} \\ {} & \circ & \textrm {Convert}\hspace {3pt} x^{m}\cdot y\hspace {3pt}\textrm {to series expansion for}\hspace {3pt} m =0..1 \\ {} & {} & x^{m}\cdot y=\moverset {\infty }{\munderset {k =0}{\sum }}a_{k} x^{k +r +m} \\ {} & \circ & \textrm {Shift index using}\hspace {3pt} k \mathrm {->}k -m \\ {} & {} & x^{m}\cdot y=\moverset {\infty }{\munderset {k =m}{\sum }}a_{k -m} x^{k +r} \\ {} & \circ & \textrm {Convert}\hspace {3pt} x^{m}\cdot y^{\prime }\hspace {3pt}\textrm {to series expansion for}\hspace {3pt} m =1..2 \\ {} & {} & x^{m}\cdot y^{\prime }=\moverset {\infty }{\munderset {k =0}{\sum }}a_{k} \left (k +r \right ) x^{k +r -1+m} \\ {} & \circ & \textrm {Shift index using}\hspace {3pt} k \mathrm {->}k +1-m \\ {} & {} & x^{m}\cdot y^{\prime }=\moverset {\infty }{\munderset {k =-1+m}{\sum }}a_{k +1-m} \left (k +1-m +r \right ) x^{k +r} \\ {} & \circ & \textrm {Convert}\hspace {3pt} x^{m}\cdot \left (\frac {d}{d x}y^{\prime }\right )\hspace {3pt}\textrm {to series expansion for}\hspace {3pt} m =2..3 \\ {} & {} & x^{m}\cdot \left (\frac {d}{d x}y^{\prime }\right )=\moverset {\infty }{\munderset {k =0}{\sum }}a_{k} \left (k +r \right ) \left (k +r -1\right ) x^{k +r -2+m} \\ {} & \circ & \textrm {Shift index using}\hspace {3pt} k \mathrm {->}k +2-m \\ {} & {} & x^{m}\cdot \left (\frac {d}{d x}y^{\prime }\right )=\moverset {\infty }{\munderset {k =-2+m}{\sum }}a_{k +2-m} \left (k +2-m +r \right ) \left (k +1-m +r \right ) x^{k +r} \\ & {} & \textrm {Rewrite ODE with series expansions}\hspace {3pt} \\ {} & {} & -a_{0} \left (\beta -r \right ) \left (\alpha -r \right ) x^{r}+\left (\moverset {\infty }{\munderset {k =1}{\sum }}\left (-a_{k} \left (\beta -k -r \right ) \left (\alpha -k -r \right )+a_{k -1} \left (b +k -1+r \right ) \left (a +k -1+r \right )\right ) x^{k +r}\right )=0 \\ \bullet & {} & a_{0}\textrm {cannot be 0 by assumption, giving the indicial equation}\hspace {3pt} \\ {} & {} & -\left (\beta -r \right ) \left (\alpha -r \right )=0 \\ \bullet & {} & \textrm {Values of r that satisfy the indicial equation}\hspace {3pt} \\ {} & {} & r \in \left \{\alpha , \beta \right \} \\ \bullet & {} & \textrm {Each term in the series must be 0, giving the recursion relation}\hspace {3pt} \\ {} & {} & a_{k -1} \left (b +k -1+r \right ) \left (a +k -1+r \right )-a_{k} \left (-\alpha +k +r \right ) \left (k +r -\beta \right )=0 \\ \bullet & {} & \textrm {Shift index using}\hspace {3pt} k \mathrm {->}k +1 \\ {} & {} & a_{k} \left (b +k +r \right ) \left (a +k +r \right )-a_{k +1} \left (-\alpha +k +1+r \right ) \left (k +1+r -\beta \right )=0 \\ \bullet & {} & \textrm {Recursion relation that defines series solution to ODE}\hspace {3pt} \\ {} & {} & a_{k +1}=\frac {a_{k} \left (b +k +r \right ) \left (a +k +r \right )}{\left (\alpha -k -1-r \right ) \left (-k -1-r +\beta \right )} \\ \bullet & {} & \textrm {Recursion relation for}\hspace {3pt} r =\alpha \\ {} & {} & a_{k +1}=\frac {a_{k} \left (b +k +\alpha \right ) \left (a +k +\alpha \right )}{\left (-k -1\right ) \left (-k -1-\alpha +\beta \right )} \\ \bullet & {} & \textrm {Solution for}\hspace {3pt} r =\alpha \\ {} & {} & \left [y=\moverset {\infty }{\munderset {k =0}{\sum }}a_{k} x^{k +\alpha }, a_{k +1}=\frac {a_{k} \left (b +k +\alpha \right ) \left (a +k +\alpha \right )}{\left (-k -1\right ) \left (-k -1-\alpha +\beta \right )}\right ] \\ \bullet & {} & \textrm {Recursion relation for}\hspace {3pt} r =\beta \\ {} & {} & a_{k +1}=\frac {a_{k} \left (b +k +\beta \right ) \left (a +k +\beta \right )}{\left (\alpha -k -1-\beta \right ) \left (-k -1\right )} \\ \bullet & {} & \textrm {Solution for}\hspace {3pt} r =\beta \\ {} & {} & \left [y=\moverset {\infty }{\munderset {k =0}{\sum }}a_{k} x^{k +\beta }, a_{k +1}=\frac {a_{k} \left (b +k +\beta \right ) \left (a +k +\beta \right )}{\left (\alpha -k -1-\beta \right ) \left (-k -1\right )}\right ] \\ \bullet & {} & \textrm {Combine solutions and rename parameters}\hspace {3pt} \\ {} & {} & \left [y=\left (\moverset {\infty }{\munderset {k =0}{\sum }}c_{k} x^{k +\alpha }\right )+\left (\moverset {\infty }{\munderset {k =0}{\sum }}d_{k} x^{k +\beta }\right ), c_{k +1}=\frac {c_{k} \left (b +k +\alpha \right ) \left (a +k +\alpha \right )}{\left (-k -1\right ) \left (-k -1-\alpha +\beta \right )}, d_{k +1}=\frac {d_{k} \left (b +k +\beta \right ) \left (a +k +\beta \right )}{\left (\alpha -k -1-\beta \right ) \left (-k -1\right )}\right ] \end {array} \]
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: 0.141 (sec). Leaf size: 86
dsolve(diff(diff(y(x),x),x) = -((a+b+1)*x+alpha+beta-1)/x/(x-1)*diff(y(x),x)-(a*b*x-alpha*beta)/x^2/(x-1)*y(x),y(x), singsol=all)
\[ y \left (x \right ) = \left (\operatorname {hypergeom}\left (\left [1-\alpha -b , 1-a -\alpha \right ], \left [1-\alpha +\beta \right ], x\right ) x^{\beta } c_{2} +\operatorname {hypergeom}\left (\left [1-b -\beta , 1-a -\beta \right ], \left [1+\alpha -\beta \right ], x\right ) x^{\alpha } c_{1} \right ) \left (x -1\right )^{1-a -\alpha -b -\beta } \]
✓ Solution by Mathematica
Time used: 0.26 (sec). Leaf size: 52
DSolve[y''[x] == -(((-(\[Alpha]*\[Beta]) + a*b*x)*y[x])/((-1 + x)*x^2)) - ((-1 + \[Alpha] + \[Beta] + (1 + a + b)*x)*y'[x])/((-1 + x)*x),y[x],x,IncludeSingularSolutions -> True]
\[ y(x)\to (-1)^{\beta } c_2 x^{\beta } \operatorname {Hypergeometric2F1}(a+\beta ,b+\beta ,-\alpha +\beta +1,x)+(-1)^{\alpha } c_1 x^{\alpha } \operatorname {Hypergeometric2F1}(a+\alpha ,b+\alpha ,\alpha -\beta +1,x) \]