Internal problem ID [10868]
Book: Handbook of exact solutions for ordinary differential equations. By Polyanin and Zaitsev.
Second edition
Section: Chapter 2, Second-Order Differential Equations. section 2.1.2-2 Equation of form
\(y''+f(x)y'+g(x)y=0\)
Problem number: 34.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _with_linear_symmetries]]
\[ \boxed {y^{\prime \prime }+\left (a \,x^{2}+b \right ) y^{\prime }+c \left (a \,x^{2}+b -c \right ) y=0} \]
✓ Solution by Maple
Time used: 0.047 (sec). Leaf size: 174
dsolve(diff(y(x),x$2)+(a*x^2+b)*diff(y(x),x)+c*(a*x^2+b-c)*y(x)=0,y(x), singsol=all)
\[ y = c_{1} \operatorname {HeunT}\left (0, -\frac {3 \sqrt {a^{2}}}{a}, \frac {a \left (b -2 c \right ) 3^{\frac {1}{3}}}{\left (a^{2}\right )^{\frac {2}{3}}}, \frac {3^{\frac {2}{3}} \left (a^{2}\right )^{\frac {1}{6}} x}{3}\right ) {\mathrm e}^{-\frac {x \left (\left (a \,x^{2}+3 b \right ) \left (a^{2}\right )^{\frac {2}{3}}+a \left (a^{2}\right )^{\frac {1}{6}} \left (a \,x^{2}+3 b -6 c \right )\right )}{6 \left (a^{2}\right )^{\frac {2}{3}}}}+c_{2} \operatorname {HeunT}\left (0, \frac {3 \sqrt {a^{2}}}{a}, \frac {a \left (b -2 c \right ) 3^{\frac {1}{3}}}{\left (a^{2}\right )^{\frac {2}{3}}}, -\frac {3^{\frac {2}{3}} \left (a^{2}\right )^{\frac {1}{6}} x}{3}\right ) {\mathrm e}^{\frac {x \left (\left (-a \,x^{2}-3 b \right ) \left (a^{2}\right )^{\frac {2}{3}}+a \left (a^{2}\right )^{\frac {1}{6}} \left (a \,x^{2}+3 b -6 c \right )\right )}{6 \left (a^{2}\right )^{\frac {2}{3}}}} \]
✓ Solution by Mathematica
Time used: 0.915 (sec). Leaf size: 46
DSolve[y''[x]+(a*x^2+b)*y'[x]+c*(a*x^2+b-c)*y[x]==0,y[x],x,IncludeSingularSolutions -> True]
\[ y(x)\to e^{-c x} \left (c_2 \int _1^xe^{-\frac {1}{3} K[1] \left (a K[1]^2+3 b-6 c\right )}dK[1]+c_1\right ) \]