60.3.323 problem 1329

Internal problem ID [11333]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 2, linear second order
Problem number : 1329
Date solved : Tuesday, January 28, 2025 at 06:01:13 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} y^{\prime \prime }&=-\frac {\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 }}{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 )} \end{align*}

✓ Solution by Maple

Time used: 1.148 (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)
 

\[ y = c_{1} \operatorname {HeunG}\left (a , q , \alpha , \beta , \gamma , \delta , x\right )+c_{2} x^{1-\gamma } \operatorname {HeunG}\left (a , q -\left (\gamma -1\right ) \left (\left (a -1\right ) \delta +\alpha +\beta -\gamma +1\right ), \beta -\gamma +1, \alpha -\gamma +1, -\gamma +2, \delta , x\right ) \]

✓ Solution by Mathematica

Time used: 0.922 (sec). Leaf size: 67

DSolve[D[y[x],{x,2}] == -(((-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)*D[y[x],x])/((-1 + x)*x*(-a + x)),y[x],x,IncludeSingularSolutions -> True]
 

\[ y(x)\to c_2 x^{1-\gamma } \text {HeunG}[a,q-(\gamma -1) ((a-1) \delta +\alpha +\beta -\gamma +1),\alpha -\gamma +1,\beta -\gamma +1,2-\gamma ,\delta ,x]+c_1 \text {HeunG}[a,q,\alpha ,\beta ,\gamma ,\delta ,x] \]