Internal problem ID [11027]
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: 203.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _with_linear_symmetries]]
\[ \boxed {x \left (a \,x^{2}+b x +1\right ) y^{\prime \prime }+\left (\alpha \,x^{2}+\beta x +\gamma \right ) y^{\prime }+\left (x n +m \right ) y=0} \]
✗ Solution by Maple
dsolve(x*(a*x^2+b*x+1)*diff(y(x),x$2)+(alpha*x^2+beta*x+gamma)*diff(y(x),x)+(n*x+m)*y(x)=0,y(x), singsol=all)
\[ \text {No solution found} \]
✓ Solution by Mathematica
Time used: 108.623 (sec). Leaf size: 524
DSolve[x*(a*x^2+b*x+1)*y''[x]+(\[Alpha]*x^2+\[Beta]*x+\[Gamma])*y'[x]+(n*x+m)*y[x]==0,y[x],x,IncludeSingularSolutions -> True]
\[ y(x)\to -\frac {2^{-\gamma } \left (-\frac {a x}{\sqrt {b^2-4 a}+b}\right )^{1-\gamma } \left (a \left (-2^{\gamma }\right ) c_1 x \left (-\frac {a x}{\sqrt {b^2-4 a}+b}\right )^{\gamma -1} \text {HeunG}\left [\frac {b-\sqrt {b^2-4 a}}{\sqrt {b^2-4 a}+b},\frac {2 m}{\sqrt {b^2-4 a}+b},\frac {1}{2} \left (-\sqrt {\frac {a^2+\alpha ^2-2 a (\alpha +2 n)}{a^2}}+\frac {\alpha }{a}-1\right ),\frac {1}{2} \left (\sqrt {\frac {a^2+\alpha ^2-2 a (\alpha +2 n)}{a^2}}+\frac {\alpha }{a}-1\right ),\gamma ,\frac {2 a^2 \gamma -a \left (\beta \left (\sqrt {b^2-4 a}+b\right )+2 \alpha \right )+\alpha b \left (\sqrt {b^2-4 a}+b\right )}{a \left (b \left (\sqrt {b^2-4 a}+b\right )-4 a\right )},-\frac {2 a x}{\sqrt {b^2-4 a}+b}\right ]-2 a c_2 x \text {HeunG}\left [\frac {b-\sqrt {b^2-4 a}}{\sqrt {b^2-4 a}+b},\frac {2 ((\gamma -1) (b \gamma -\beta )+m)}{\sqrt {b^2-4 a}+b},\frac {1}{2} \left (-\sqrt {\frac {a^2+\alpha ^2-2 a (\alpha +2 n)}{a^2}}+\frac {\alpha }{a}-2 \gamma +1\right ),\frac {1}{2} \left (\sqrt {\frac {a^2+\alpha ^2-2 a (\alpha +2 n)}{a^2}}+\frac {\alpha }{a}-2 \gamma +1\right ),2-\gamma ,\frac {2 a^2 \gamma -a \left (\beta \left (\sqrt {b^2-4 a}+b\right )+2 \alpha \right )+\alpha b \left (\sqrt {b^2-4 a}+b\right )}{a \left (b \left (\sqrt {b^2-4 a}+b\right )-4 a\right )},-\frac {2 a x}{\sqrt {b^2-4 a}+b}\right ]\right )}{a x} \]