Internal problem ID [11022]
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: 198.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _with_linear_symmetries]]
\[ \boxed {\left (a \,x^{3}+b \,x^{2}+c x \right ) y^{\prime \prime }+\left (n \,x^{2}+m x +k \right ) y^{\prime }+\left (k -1\right ) \left (\left (-a k +n \right ) x +m -b k \right ) y=0} \]
✗ Solution by Maple
dsolve((a*x^3+b*x^2+c*x)*diff(y(x),x$2)+(n*x^2+m*x+k)*diff(y(x),x)+(k-1)*((n-a*k)*x+m-b*k)*y(x)=0,y(x), singsol=all)
\[ \text {No solution found} \]
✓ Solution by Mathematica
Time used: 148.451 (sec). Leaf size: 570
DSolve[(a*x^3+b*x^2+c*x)*y''[x]+(n*x^2+m*x+k)*y'[x]+(k-1)*((n-a*k)*x+m-b*k)*y[x]==0,y[x],x,IncludeSingularSolutions -> True]
\[ y(x)\to -\frac {2^{-\frac {k}{c}} \left (-\frac {a x}{\sqrt {b^2-4 a c}+b}\right )^{1-\frac {k}{c}} \left (a c_1 x \left (-2^{\frac {k}{c}}\right ) \left (-\frac {a x}{\sqrt {b^2-4 a c}+b}\right )^{\frac {k}{c}-1} \text {HeunG}\left [\frac {b-\sqrt {b^2-4 a c}}{\sqrt {b^2-4 a c}+b},-\frac {2 (k-1) (b k-m)}{\sqrt {b^2-4 a c}+b},\frac {1}{2} \left (-\sqrt {\frac {(-2 a k+a+n)^2}{a^2}}+\frac {n}{a}-1\right ),\frac {1}{2} \left (\sqrt {\frac {(-2 a k+a+n)^2}{a^2}}+\frac {n}{a}-1\right ),\frac {k}{c},\frac {2 a^2 k-a \left (m \sqrt {b^2-4 a c}+b m+2 c n\right )+b n \left (\sqrt {b^2-4 a c}+b\right )}{a \left (b \sqrt {b^2-4 a c}-4 a c+b^2\right )},-\frac {2 a x}{\sqrt {b^2-4 a c}+b}\right ]-2 a c_2 x \text {HeunG}\left [\frac {b-\sqrt {b^2-4 a c}}{\sqrt {b^2-4 a c}+b},-\frac {2 (c-1) k (b (c (k-1)+k)-c m)}{c^2 \left (\sqrt {b^2-4 a c}+b\right )},\frac {1}{2} \left (-\sqrt {\frac {(-2 a k+a+n)^2}{a^2}}+\frac {n}{a}-\frac {2 k}{c}+1\right ),\frac {a \sqrt {\frac {(-2 a k+a+n)^2}{a^2}}+a+n}{2 a}-\frac {k}{c},2-\frac {k}{c},\frac {2 a^2 k-a \left (m \sqrt {b^2-4 a c}+b m+2 c n\right )+b n \left (\sqrt {b^2-4 a c}+b\right )}{a \left (b \sqrt {b^2-4 a c}-4 a c+b^2\right )},-\frac {2 a x}{\sqrt {b^2-4 a c}+b}\right ]\right )}{a x} \]