Internal problem ID [11038]
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 Equation of form
\((a_3 x^3+a_2 x^2 x+a_1 x+a_0) y''+f(x)y'+g(x)y=0\)
Problem number: 204.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _with_linear_symmetries]]
\[ \boxed {x \left (x -1\right ) \left (x -a \right ) y^{\prime \prime }+\left (\left (\beta +\alpha +1\right ) x^{2}-\left (\alpha +\beta +1+a \left (\gamma +d \right )-a \right ) x +a \gamma \right ) y^{\prime }+\left (\alpha \beta x -q \right ) y=0} \]
✓ Solution by Maple
Time used: 0.141 (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)
\[ y \left (x \right ) = c_{1} \operatorname {HeunG}\left (a , q , \alpha , \beta , \gamma , \frac {a \left (d -1\right )}{a -1}, x\right )+c_{2} x^{1-\gamma } \operatorname {HeunG}\left (a , q -\left (-1+\gamma \right ) \left (a \left (d -1\right )+\alpha +\beta -\gamma +1\right ), \beta +1-\gamma , \alpha +1-\gamma , 2-\gamma , \frac {a \left (d -1\right )}{a -1}, x\right ) \]
✓ Solution by Mathematica
Time used: 2.215 (sec). Leaf size: 85
DSolve[x*(x-1)*(x-a)*y''[x]+((\[Alpha]+\[Beta]+1)*x^2-(\[Alpha]+\[Beta]+1+a*(\[Gamma]+d)-a)*x+a*\[Gamma])*y'[x]+(\[Alpha]*\[Beta]*x-q)*y[x]==0,y[x],x,IncludeSingularSolutions -> True]
\[ y(x)\to c_2 x^{1-\gamma } \text {HeunG}\left [a,q-(\gamma -1) (a (d-1)+\alpha +\beta -\gamma +1),\alpha -\gamma +1,\beta -\gamma +1,2-\gamma ,\frac {a (d-1)}{a-1},x\right ]+c_1 \text {HeunG}\left [a,q,\alpha ,\beta ,\gamma ,\frac {a (d-1)}{a-1},x\right ] \]