Internal problem ID [9656]
Internal file name [OUTPUT/8598_Monday_June_06_2022_04_20_44_AM_69220225/index.tex]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 2, linear second order
Problem number: 1329.
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.
\[ \boxed {y^{\prime \prime }+\frac {\left (\left (\beta +\alpha +1\right ) x^{2}-\left (\alpha +\beta +1+a \left (\gamma +\delta \right )-\delta \right ) x +a \gamma \right ) y^{\prime }}{x \left (x -1\right ) \left (x -a \right )}+\frac {\left (\alpha \beta x -q \right ) y}{x \left (x -1\right ) \left (x -a \right )}=0} \]
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & \left (\frac {d}{d x}y^{\prime }\right ) x \left (x -1\right ) \left (-x +a \right )+\left (\left (-\alpha -\beta -1\right ) x^{2}+\left (\alpha +\beta +1+a \left (\gamma +\delta \right )-\delta \right ) x -a \gamma \right ) y^{\prime }+\left (-\alpha \beta x +q \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 (\alpha \beta x -q \right ) y}{x \left (x -1\right ) \left (-x +a \right )}-\frac {\left (a \delta x +a \gamma x -x^{2} \alpha -\beta \,x^{2}-a \gamma +\alpha x +\beta x -\delta x -x^{2}+x \right ) y^{\prime }}{x \left (x -1\right ) \left (-x +a \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 (a \delta x +a \gamma x -x^{2} \alpha -\beta \,x^{2}-a \gamma +\alpha x +\beta x -\delta x -x^{2}+x \right ) y^{\prime }}{x \left (x -1\right ) \left (-x +a \right )}-\frac {\left (\alpha \beta x -q \right ) y}{x \left (x -1\right ) \left (-x +a \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 {a \delta x +a \gamma x -x^{2} \alpha -\beta \,x^{2}-a \gamma +\alpha x +\beta x -\delta x -x^{2}+x}{x \left (x -1\right ) \left (-x +a \right )}, P_{3}\left (x \right )=-\frac {\alpha \beta x -q}{x \left (x -1\right ) \left (-x +a \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}}}=\gamma \\ {} & \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}}}=0 \\ {} & \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 \left (x -1\right ) \left (-x +a \right )+\left (a \delta x +a \gamma x -x^{2} \alpha -\beta \,x^{2}-a \gamma +\alpha x +\beta x -\delta x -x^{2}+x \right ) y^{\prime }+\left (-\alpha \beta x +q \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 =0..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 =1..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 a_{0} r \left (-1+r +\gamma \right ) x^{-1+r}+\left (-a a_{1} \left (1+r \right ) \left (r +\gamma \right )+a_{0} \left (a \delta r +a \gamma r +a \,r^{2}-a r +\alpha r +\beta r -\delta r +r^{2}+q \right )\right ) x^{r}+\left (\moverset {\infty }{\munderset {k =1}{\sum }}\left (-a a_{k +1} \left (k +1+r \right ) \left (k +r +\gamma \right )+a_{k} \left (a \delta k +a \delta r +a \gamma k +a \gamma r +a \,k^{2}+2 a k r +a \,r^{2}-a k -a r +\alpha k +\alpha r +\beta k +\beta r -\delta k -\delta r +k^{2}+2 k r +r^{2}+q \right )-a_{k -1} \left (\beta +k -1+r \right ) \left (\alpha +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} \\ {} & {} & -a r \left (-1+r +\gamma \right )=0 \\ \bullet & {} & \textrm {Values of r that satisfy the indicial equation}\hspace {3pt} \\ {} & {} & r \in \left \{0, 1-\gamma \right \} \\ \bullet & {} & \textrm {Each term must be 0}\hspace {3pt} \\ {} & {} & -a a_{1} \left (1+r \right ) \left (r +\gamma \right )+a_{0} \left (a \delta r +a \gamma r +a \,r^{2}-a r +\alpha r +\beta r -\delta r +r^{2}+q \right )=0 \\ \bullet & {} & \textrm {Each term in the series must be 0, giving the recursion relation}\hspace {3pt} \\ {} & {} & -a a_{k +1} \left (k +1+r \right ) \left (k +r +\gamma \right )+a_{k} \left (\left (k +r \right ) \left (k +r +\delta +\gamma -1\right ) a +\left (-k -r \right ) \delta +r^{2}+\left (2 k +\beta +\alpha \right ) r +k^{2}+\left (\beta +\alpha \right ) k +q \right )-a_{k -1} \left (\beta +k -1+r \right ) \left (\alpha +k -1+r \right )=0 \\ \bullet & {} & \textrm {Shift index using}\hspace {3pt} k \mathrm {->}k +1 \\ {} & {} & -a a_{k +2} \left (k +2+r \right ) \left (k +1+r +\gamma \right )+a_{k +1} \left (\left (k +1+r \right ) \left (k +r +\delta +\gamma \right ) a +\left (-k -1-r \right ) \delta +r^{2}+\left (2 k +2+\beta +\alpha \right ) r +\left (k +1\right )^{2}+\left (\beta +\alpha \right ) \left (k +1\right )+q \right )-a_{k} \left (\beta +k +r \right ) \left (\alpha +k +r \right )=0 \\ \bullet & {} & \textrm {Recursion relation that defines series solution to ODE}\hspace {3pt} \\ {} & {} & a_{k +2}=\frac {a \delta k a_{k +1}+a \delta r a_{k +1}+a \gamma k a_{k +1}+a \gamma r a_{k +1}+a \,k^{2} a_{k +1}+2 a k r a_{k +1}+a \,r^{2} a_{k +1}+a \delta a_{k +1}+a \gamma a_{k +1}+a k a_{k +1}+a r a_{k +1}-a_{k} \alpha \beta -a_{k} \alpha k +\alpha k a_{k +1}-\alpha r a_{k}+\alpha r a_{k +1}-a_{k} \beta k +\beta k a_{k +1}-\beta r a_{k}+\beta r a_{k +1}-\delta k a_{k +1}-\delta r a_{k +1}-k^{2} a_{k}+k^{2} a_{k +1}-2 k r a_{k}+2 k r a_{k +1}-r^{2} a_{k}+r^{2} a_{k +1}+\alpha a_{k +1}+\beta a_{k +1}-\delta a_{k +1}+2 k a_{k +1}+q a_{k +1}+2 r a_{k +1}+a_{k +1}}{a \left (k +2+r \right ) \left (k +1+r +\gamma \right )} \\ \bullet & {} & \textrm {Recursion relation for}\hspace {3pt} r =0 \\ {} & {} & a_{k +2}=\frac {a \delta k a_{k +1}+a \gamma k a_{k +1}+a \,k^{2} a_{k +1}+a \delta a_{k +1}+a \gamma a_{k +1}+a k a_{k +1}-a_{k} \alpha \beta -a_{k} \alpha k +\alpha k a_{k +1}-a_{k} \beta k +\beta k a_{k +1}-\delta k a_{k +1}-k^{2} a_{k}+k^{2} a_{k +1}+\alpha a_{k +1}+\beta a_{k +1}-\delta a_{k +1}+2 k a_{k +1}+q a_{k +1}+a_{k +1}}{a \left (k +2\right ) \left (k +1+\gamma \right )} \\ \bullet & {} & \textrm {Solution for}\hspace {3pt} r =0 \\ {} & {} & \left [y=\moverset {\infty }{\munderset {k =0}{\sum }}a_{k} x^{k}, a_{k +2}=\frac {a \delta k a_{k +1}+a \gamma k a_{k +1}+a \,k^{2} a_{k +1}+a \delta a_{k +1}+a \gamma a_{k +1}+a k a_{k +1}-a_{k} \alpha \beta -a_{k} \alpha k +\alpha k a_{k +1}-a_{k} \beta k +\beta k a_{k +1}-\delta k a_{k +1}-k^{2} a_{k}+k^{2} a_{k +1}+\alpha a_{k +1}+\beta a_{k +1}-\delta a_{k +1}+2 k a_{k +1}+q a_{k +1}+a_{k +1}}{a \left (k +2\right ) \left (k +1+\gamma \right )}, -a \gamma a_{1}+q a_{0}=0\right ] \\ \bullet & {} & \textrm {Recursion relation for}\hspace {3pt} r =1-\gamma \\ {} & {} & a_{k +2}=\frac {a \gamma k a_{k +1}+a \delta \left (1-\gamma \right ) a_{k +1}+a \gamma \left (1-\gamma \right ) a_{k +1}+2 a k \left (1-\gamma \right ) a_{k +1}-a_{k} \alpha \beta -a_{k} \alpha k -a_{k} \beta k +a \,k^{2} a_{k +1}+a \gamma a_{k +1}+a k a_{k +1}+a \left (1-\gamma \right )^{2} a_{k +1}+a \left (1-\gamma \right ) a_{k +1}-a_{k} \alpha \left (1-\gamma \right )+\alpha \left (1-\gamma \right ) a_{k +1}-a_{k} \beta \left (1-\gamma \right )+\beta \left (1-\gamma \right ) a_{k +1}-\delta \left (1-\gamma \right ) a_{k +1}-2 k \left (1-\gamma \right ) a_{k}+2 k \left (1-\gamma \right ) a_{k +1}+\beta a_{k +1}-\delta a_{k +1}+2 k a_{k +1}+q a_{k +1}+k^{2} a_{k +1}+\alpha a_{k +1}+a \delta k a_{k +1}+a_{k +1}+a \delta a_{k +1}+\alpha k a_{k +1}+\beta k a_{k +1}-\delta k a_{k +1}-k^{2} a_{k}-\left (1-\gamma \right )^{2} a_{k}+\left (1-\gamma \right )^{2} a_{k +1}+2 \left (1-\gamma \right ) a_{k +1}}{a \left (k +3-\gamma \right ) \left (k +2\right )} \\ \bullet & {} & \textrm {Solution for}\hspace {3pt} r =1-\gamma \\ {} & {} & \left [y=\moverset {\infty }{\munderset {k =0}{\sum }}a_{k} x^{-\gamma +k +1}, a_{k +2}=\frac {a \gamma k a_{k +1}+a \delta \left (1-\gamma \right ) a_{k +1}+a \gamma \left (1-\gamma \right ) a_{k +1}+2 a k \left (1-\gamma \right ) a_{k +1}-a_{k} \alpha \beta -a_{k} \alpha k -a_{k} \beta k +a \,k^{2} a_{k +1}+a \gamma a_{k +1}+a k a_{k +1}+a \left (1-\gamma \right )^{2} a_{k +1}+a \left (1-\gamma \right ) a_{k +1}-a_{k} \alpha \left (1-\gamma \right )+\alpha \left (1-\gamma \right ) a_{k +1}-a_{k} \beta \left (1-\gamma \right )+\beta \left (1-\gamma \right ) a_{k +1}-\delta \left (1-\gamma \right ) a_{k +1}-2 k \left (1-\gamma \right ) a_{k}+2 k \left (1-\gamma \right ) a_{k +1}+\beta a_{k +1}-\delta a_{k +1}+2 k a_{k +1}+q a_{k +1}+k^{2} a_{k +1}+\alpha a_{k +1}+a \delta k a_{k +1}+a_{k +1}+a \delta a_{k +1}+\alpha k a_{k +1}+\beta k a_{k +1}-\delta k a_{k +1}-k^{2} a_{k}-\left (1-\gamma \right )^{2} a_{k}+\left (1-\gamma \right )^{2} a_{k +1}+2 \left (1-\gamma \right ) a_{k +1}}{a \left (k +3-\gamma \right ) \left (k +2\right )}, -a a_{1} \left (-\gamma +2\right )+a_{0} \left (a \delta \left (1-\gamma \right )+a \gamma \left (1-\gamma \right )+a \left (1-\gamma \right )^{2}-a \left (1-\gamma \right )+\alpha \left (1-\gamma \right )+\beta \left (1-\gamma \right )-\left (1-\gamma \right ) \delta +\left (1-\gamma \right )^{2}+q \right )=0\right ] \\ \bullet & {} & \textrm {Combine solutions and rename parameters}\hspace {3pt} \\ {} & {} & \left [y=\left (\moverset {\infty }{\munderset {k =0}{\sum }}b_{k} x^{k}\right )+\left (\moverset {\infty }{\munderset {k =0}{\sum }}c_{k} x^{-\gamma +k +1}\right ), b_{k +2}=\frac {a \delta k b_{k +1}+a \gamma k b_{k +1}+a \,k^{2} b_{k +1}+a \delta b_{k +1}+a \gamma b_{k +1}+a k b_{k +1}-\alpha \beta b_{k}-\alpha k b_{k}+\alpha k b_{k +1}-\beta k b_{k}+\beta k b_{k +1}-\delta k b_{k +1}-k^{2} b_{k}+k^{2} b_{k +1}+\alpha b_{k +1}+\beta b_{k +1}-\delta b_{k +1}+2 k b_{k +1}+q b_{k +1}+b_{k +1}}{a \left (k +2\right ) \left (k +1+\gamma \right )}, -a \gamma b_{1}+q b_{0}=0, c_{k +2}=\frac {a \gamma k c_{k +1}+a \delta \left (1-\gamma \right ) c_{k +1}+a \gamma \left (1-\gamma \right ) c_{k +1}+2 a k \left (1-\gamma \right ) c_{k +1}+a \delta k c_{k +1}+\beta c_{k +1}-\delta c_{k +1}+2 k c_{k +1}+q c_{k +1}+k^{2} c_{k +1}+\alpha c_{k +1}-k^{2} c_{k}-\left (1-\gamma \right )^{2} c_{k}+\left (1-\gamma \right )^{2} c_{k +1}+2 \left (1-\gamma \right ) c_{k +1}+c_{k +1}-c_{k} \alpha \beta -c_{k} \alpha k -c_{k} \beta k +a \,k^{2} c_{k +1}+a \gamma c_{k +1}+a k c_{k +1}+a \left (1-\gamma \right )^{2} c_{k +1}+a \left (1-\gamma \right ) c_{k +1}-c_{k} \alpha \left (1-\gamma \right )+\alpha \left (1-\gamma \right ) c_{k +1}-c_{k} \beta \left (1-\gamma \right )+\beta \left (1-\gamma \right ) c_{k +1}-\delta \left (1-\gamma \right ) c_{k +1}-2 k \left (1-\gamma \right ) c_{k}+2 k \left (1-\gamma \right ) c_{k +1}+a \delta c_{k +1}+\alpha k c_{k +1}+\beta k c_{k +1}-\delta k c_{k +1}}{a \left (k +3-\gamma \right ) \left (k +2\right )}, -a c_{1} \left (-\gamma +2\right )+c_{0} \left (a \delta \left (1-\gamma \right )+a \gamma \left (1-\gamma \right )+a \left (1-\gamma \right )^{2}-a \left (1-\gamma \right )+\alpha \left (1-\gamma \right )+\beta \left (1-\gamma \right )-\left (1-\gamma \right ) \delta +\left (1-\gamma \right )^{2}+q \right )=0\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 -> hyper3: Equivalence to 2F1, 1F1 or 0F1 under a power @ Moebius -> Mathieu -> Equivalence to the rational form of Mathieu ODE under a power @ Moebius trying a solution in terms of MeijerG functions -> Heun: Equivalence to the GHE or one of its 4 confluent cases under a power @ Moebius <- Heun successful: received ODE is equivalent to the HeunG ODE, case a <> 0, e <> 0, g <> 0, c = 0 `
✓ Solution by Maple
Time used: 0.765 (sec). Leaf size: 64
dsolve(diff(diff(y(x),x),x) = -((alpha+beta+1)*x^2-(alpha+beta+1+a*(gamma+delta)-delta)*x+a*gamma)/x/(x-1)/(x-a)*diff(y(x),x)-(alpha*beta*x-q)/x/(x-1)/(x-a)*y(x),y(x), singsol=all)
\[ y \left (x \right ) = c_{1} \operatorname {HeunG}\left (a , q , \alpha , \beta , \gamma , \delta , x\right )+c_{2} x^{1-\gamma } \operatorname {HeunG}\left (a , q -\left (-1+\gamma \right ) \left (\delta \left (a -1\right )+\alpha +\beta -\gamma +1\right ), \beta +1-\gamma , \alpha +1-\gamma , -\gamma +2, \delta , x\right ) \]
✓ Solution by Mathematica
Time used: 0.927 (sec). Leaf size: 67
DSolve[y''[x] == -(((-q + \[Alpha]*\[Beta]*x)*y[x])/((-1 + x)*x*(-a + x))) - ((a*\[Gamma] - (1 + \[Alpha] +\[Beta] - \[Delta] + a*(\[Delta] + \[Gamma]))*x + (1 + \[Alpha] + \[Beta])*x^2)*y'[x])/((-1 + x)*x*(-a + x)),y[x],x,IncludeSingularSolutions -> True]
\[ y(x)\to c_2 x^{1-\gamma } \text {HeunG}[a,q-(\gamma -1) ((a-1) \delta +\alpha +\beta -\gamma +1),\alpha -\gamma +1,\beta -\gamma +1,2-\gamma ,\delta ,x]+c_1 \text {HeunG}[a,q,\alpha ,\beta ,\gamma ,\delta ,x] \]