Internal problem ID [11034]
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: 210.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _with_linear_symmetries]]
\[ \boxed {2 x \left (a \,x^{2}+b x +c \right ) y^{\prime \prime }+\left (a \left (2-k \right ) x^{2}+b \left (-k +1\right ) x -c k \right ) y^{\prime }+\lambda \,x^{1+k} y=0} \]
✗ Solution by Maple
dsolve(2*x(a*x^2+b*x+c)*diff(y(x),x$2)+(a*(2-k)*x^2+b*(1-k)*x-c*k)*diff(y(x),x)+(lambda*x^(k+1))*y(x)=0,y(x), singsol=all)
\[ \text {No solution found} \]
✓ Solution by Mathematica
Time used: 157.344 (sec). Leaf size: 790
DSolve[2*x(a*x^2+b*x+c)*y''[x]+(a*(2-k)*x^2+b*(1-k)*x-c*k)*y'[x]+(\[Lambda]*x^(k+1))*y[x]==0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \frac {\sqrt {2} \sqrt {c_1} \tan \left (\frac {\sqrt {2} x \sqrt {\frac {-\sqrt {b^2-4 a c}+2 a x+b}{b-\sqrt {b^2-4 a c}}} \sqrt {\frac {\sqrt {b^2-4 a c}+2 a x+b}{\sqrt {b^2-4 a c}+b}} \operatorname {AppellF1}\left (\frac {k+2}{2},\frac {1}{2},\frac {1}{2},\frac {k+4}{2},-\frac {2 a x}{b+\sqrt {b^2-4 a c}},\frac {2 a x}{\sqrt {b^2-4 a c}-b}\right )}{(k+2) \sqrt {\frac {x^{-k} (x (a x+b)+c)}{\lambda }}}-c_2\right )}{\sqrt {-1-\tan ^2\left (\frac {\sqrt {2} x \sqrt {\frac {-\sqrt {b^2-4 a c}+2 a x+b}{b-\sqrt {b^2-4 a c}}} \sqrt {\frac {\sqrt {b^2-4 a c}+2 a x+b}{\sqrt {b^2-4 a c}+b}} \operatorname {AppellF1}\left (\frac {k+2}{2},\frac {1}{2},\frac {1}{2},\frac {k+4}{2},-\frac {2 a x}{b+\sqrt {b^2-4 a c}},\frac {2 a x}{\sqrt {b^2-4 a c}-b}\right )}{(k+2) \sqrt {\frac {x^{-k} (x (a x+b)+c)}{\lambda }}}-c_2\right )}} \\ y(x)\to -\frac {\sqrt {2} \sqrt {c_1} \tan \left (\frac {\sqrt {2} x \sqrt {\frac {-\sqrt {b^2-4 a c}+2 a x+b}{b-\sqrt {b^2-4 a c}}} \sqrt {\frac {\sqrt {b^2-4 a c}+2 a x+b}{\sqrt {b^2-4 a c}+b}} \operatorname {AppellF1}\left (\frac {k+2}{2},\frac {1}{2},\frac {1}{2},\frac {k+4}{2},-\frac {2 a x}{b+\sqrt {b^2-4 a c}},\frac {2 a x}{\sqrt {b^2-4 a c}-b}\right )}{(k+2) \sqrt {\frac {x^{-k} (x (a x+b)+c)}{\lambda }}}+c_2\right )}{\sqrt {-1-\tan ^2\left (\frac {\sqrt {2} x \sqrt {\frac {-\sqrt {b^2-4 a c}+2 a x+b}{b-\sqrt {b^2-4 a c}}} \sqrt {\frac {\sqrt {b^2-4 a c}+2 a x+b}{\sqrt {b^2-4 a c}+b}} \operatorname {AppellF1}\left (\frac {k+2}{2},\frac {1}{2},\frac {1}{2},\frac {k+4}{2},-\frac {2 a x}{b+\sqrt {b^2-4 a c}},\frac {2 a x}{\sqrt {b^2-4 a c}-b}\right )}{(k+2) \sqrt {\frac {x^{-k} (x (a x+b)+c)}{\lambda }}}+c_2\right )}} \\ \end{align*}