Internal problem ID [8908]
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.
CAS Maple gives this as type [[_2nd_order, _with_linear_symmetries]]
Solve \begin {gather*} \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} \end {gather*}
✓ Solution by Maple
Time used: 0.593 (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)
\[ y \relax (x ) = c_{1} \mathit {HG}\left (a , q , \alpha , \beta , \gamma , \delta , x\right )+c_{2} x^{1-\gamma } \mathit {HG}\left (a , q -\left (-1+\gamma \right ) \left (\delta \left (a -1\right )+\alpha +\beta -\gamma +1\right ), \beta +1-\gamma , \alpha +1-\gamma , -\gamma +2, \delta , x\right ) \]
✓ Solution by Mathematica
Time used: 0.43 (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]
\begin{align*} 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] \\ \end{align*}