61.31.23 problem 204

Internal problem ID [12704]
Book : Handbook of exact solutions for ordinary differential equations. By Polyanin and Zaitsev. Second edition
Section : Chapter 2, Second-Order Differential Equations. section 2.1.2-6
Problem number : 204
Date solved : Tuesday, January 28, 2025 at 08:11:32 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} x \left (x -1\right ) \left (x -a \right ) y^{\prime \prime }+\left (\left (\alpha +\beta +1\right ) x^{2}-\left (\alpha +\beta +1+a \left (\gamma +d \right )-a \right ) x +a \gamma \right ) y^{\prime }+\left (\alpha \beta x -q \right ) y&=0 \end{align*}

✓ Solution by Maple

Time used: 1.837 (sec). Leaf size: 82

dsolve(x*(x-1)*(x-a)*diff(y(x),x$2)+((alpha+beta+1)*x^2-(alpha+beta+1+a*(gamma+d)-a)*x+a*gamma)*diff(y(x),x)+(alpha*beta*x-q)*y(x)=0,y(x), singsol=all)
 

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

✓ Solution by Mathematica

Time used: 1.321 (sec). Leaf size: 85

DSolve[x*(x-1)*(x-a)*D[y[x],{x,2}]+((\[Alpha]+\[Beta]+1)*x^2-(\[Alpha]+\[Beta]+1+a*(\[Gamma]+d)-a)*x+a*\[Gamma])*D[y[x],x]+(\[Alpha]*\[Beta]*x-q)*y[x]==0,y[x],x,IncludeSingularSolutions -> True]
 

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