Internal problem ID [10873]
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: 39.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _with_linear_symmetries]]
\[ \boxed {y^{\prime \prime }+\left (a \,x^{2}+b x +c \right ) y^{\prime }+x \left (b a \,x^{2}+b c +2 a \right ) y=0} \]
✓ Solution by Maple
Time used: 0.047 (sec). Leaf size: 240
dsolve(diff(y(x),x$2)+(a*x^2+b*x+c)*diff(y(x),x)+x*(a*b*x^2+b*c+2*a)*y(x)=0,y(x), singsol=all)
\[ y = c_{1} {\mathrm e}^{-\frac {\left (\sqrt {a^{2}}+a \right ) \sqrt {a^{2}}\, x^{3}}{6 a}-\frac {b \left (\sqrt {a^{2}}-a \right ) \sqrt {a^{2}}\, x^{2}}{4 a^{2}}-\frac {c \left (\sqrt {a^{2}}+a \right ) \sqrt {a^{2}}\, x}{2 a^{2}}} \operatorname {HeunT}\left (0, \frac {3 \sqrt {a^{2}}}{a}, \frac {3^{\frac {1}{3}} \left (4 a c -b^{2}\right )}{4 \left (a^{2}\right )^{\frac {2}{3}}}, \frac {3^{\frac {2}{3}} a \left (2 a x -b \right )}{6 \left (a^{2}\right )^{\frac {5}{6}}}\right )+c_{2} {\mathrm e}^{-\frac {\left (\sqrt {a^{2}}-a \right ) \sqrt {a^{2}}\, x^{3}}{6 a}-\frac {b \left (\sqrt {a^{2}}+a \right ) \sqrt {a^{2}}\, x^{2}}{4 a^{2}}-\frac {c \left (\sqrt {a^{2}}-a \right ) \sqrt {a^{2}}\, x}{2 a^{2}}} \operatorname {HeunT}\left (0, -\frac {3 \sqrt {a^{2}}}{a}, \frac {3^{\frac {1}{3}} \left (4 a c -b^{2}\right )}{4 \left (a^{2}\right )^{\frac {2}{3}}}, -\frac {\left (a x -\frac {b}{2}\right ) a 3^{\frac {2}{3}}}{3 \left (a^{2}\right )^{\frac {5}{6}}}\right ) \]
✓ Solution by Mathematica
Time used: 1.085 (sec). Leaf size: 59
DSolve[y''[x]+(a*x^2+b*x+c)*y'[x]+x*(a*b*x^2+b*c+2*a)*y[x]==0,y[x],x,IncludeSingularSolutions -> True]
\[ y(x)\to e^{-\frac {1}{3} x \left (a x^2+3 c\right )} \left (c_2 \int _1^x\exp \left (\frac {1}{6} K[1] (6 c+K[1] (2 a K[1]-3 b))\right )dK[1]+c_1\right ) \]